H 9THETRACEFORMULAbarrett/resources/obarrett... · 2017. 12. 24. · of any C2(Σg) function...

THE SPECTRUM OF THE LAPLACIAN ON ΓH THE TRACE FORMULA

OWEN BARRETT

Abstract Maaszlig cusp forms on PSL(2Z)H are the basic building blocks of automor-phic forms on GL(2) Yet their existence is known almost entirely through indirect meansno one has found a single example of an everywhere-unramified cuspidal Maaszlig form forPSL(2Z) The spectral theory of PSL(2Z)H is deeply important and remains exceedinglymysterious In this expository article we introduce some of the key questions in this areaand the main tool used to study them the Selberg trace formula

Contents

1 Introduction 22 The Trace Formula Compact Case 63 The Trace Formula Arithmetic Case 12

31 Introducing the Selberg Trace Formula on ΓH 1432 The Spectral Decomposition of ΓH 1733 The Selberg Trace Formula on ΓH 25

4 The Spectrum in Detail 3641 The Continuous Spectrum Eisenstein series on ΓH 3642 The Discrete Cuspidal Spectrum Maaszlig forms on ΓH 4043 The Ramanujan Conjecture 5044 Weylrsquos Law 5145 Quantum Ergodicity 5446 Dukersquos Theorem 56

References 59

Elle est retrouveacuteeQuoi LrsquoEacuteterniteacuteCrsquoest la mer alleacuteeAvec le soleilArthur Rimbaud

Date 27 April 2015The author would like to thank Peng Zhao who supervised this essay for many generous conversations

1

2 OWEN BARRETT

1 Introduction

The goal of this exposition is to discuss the spectrum of the Laplace-Beltrami operator Δon (compact and arithmetic) manifolds given by quotients of the upper half-planeH = z isinC imagez gt 0 and to introduce themain tool available to study the structure of the spectrumthe Selberg trace formula

On a compact hyperbolic surface S Δ can be interpreted as the quantum Hamiltonian ofa particle whose classical dynamics are described by the (chaotic) geodesic flow on the unittangent bundle of S The spectrum of Δ on S resists straightforward characterization andindeed it is only by considering the spectrum all at once that Selberg was able to find theidentity that bears his name He called this identity a lsquotrace formularsquo since it was obtainedby computing the trace of an integral operator on S of trace class in two different ways Thismethod generalizes the Poisson summation formula which is obtained in precisely the sameway In every case in which a trace formula exists the formula exploits the duality affordedby Fourier theory to relate on average a collection of spectral data to a collection of geo-metric data

An understanding of the spectrum and eigenfunctions of the Laplacian on a compactRiemannianmanifold S is of interest to physicists For example a basic problem that probesthe interaction between classical and quantum mechanics is the question of whether or notthe L2 mass of high-energy eigenfunctions becomes equidistributed over S in the limit Itis known that when the geodesic flow on S (corresponding to the classical dynamics) is er-godic such equidistribution is achieved generically we call such equidistribution quantumergodicity We would also like to know whether the genericity hypothesis may be lifted ifnot there would be a subset of lsquobadrsquo eigenfunctions that resist equidistribution If we needno assumption on genericity and all eigenfunctions equidistribute in the limit of high en-ergy we say the system exhibits quantum unique ergodicity This phenomenon and relatedconjectures are discussed in sect45

Automorphic forms and their associated L-functions are principal objects of study bymany contemporary number theorists Perhaps the most classical motivation may be foundin the connection between modular forms and elliptic integrals or modular functionsrsquo re-lationship to generating functions for combinatorial and number-theoretic quantities ofwhich the connection between the partition function and the Dedekind eta function is butone example More recently connections have been made between the coefficients of cer-tain modular forms and the representation theory of some sporadic groups In contempo-rary number theory modularity of elliptic curves has connected holomorphic cusp formsto elliptic curves and to the theory of Galois representations The automorphy of Galoisrepresentations or how to obtain a Galois representation from an automorphic form arequestions that are of central interest to number theorists today Statements relating to spe-cial values ofmotivic L-functions some of which are theorems likeDirichletrsquos class numberformula most of which are conjectures among them the Birch and Swinnerton-Dyer con-jecture and more generally the equivariant Tamagawa conjecture have provided clear evi-dence that global information of the utmost importance is encoded in L-functions attached

THE SPECTRUM OF THE LAPLACIAN ON ΓH amp THE TRACE FORMULA 3

to arithmetic objects Langlandsrsquos conjectures among them (a) the functoriality conjec-ture which loosely speaking posits the existence of a kind of functorial relationship be-tween automorphic forms and automorphic L-functions (operations on L-functions shouldcorrespond to operations [lifts] on automorphic forms and vice versa) (b) conjectures ex-tending endoscopic transfer of automorphic forms to more exotic sorts of transfer and (c)conjectures that suggest that all L-functions ofmotivic origin should factor as products of L-functions belonging to a certain class of lsquostandardrsquo automorphic L-functions all contributeto making the study of automorphic forms representations and L-functions a major focusof contemporary number theory

The majority of meaningful statements one could hope to make about L-functions at-tached to automorphic forms such as subconvexity bounds distributional statistics of crit-ical zeros and in particular low-lying zeros bounds on residues moments bounds and av-erage values at the central point all require statements about spectral properties of the au-tomorphic forms themselves On the other hand spectral statements about automorphicforms and related objects have been found to have striking and ingenious applications tostatements as seemingly unrelated as equidistribution of geodesics on the modular surface(cf sect46) Trace formulaelig are the primary route to obtain necessary spectral insight and arecrucial ingredients in obtaining statements like the above Trace formulaelig are also crucialfor understanding finer points about the decomposition of the spectrum of the Laplacianon arithmetic manifolds such as Weylrsquos law

For these reasons wewill focusmost of our attention in this article on the arithmeticman-ifold ΓH where Γ = PSL(2Z) Themodular surface ΓH is not compact and therefore thespectrum of the Laplace-Beltrami operator admits a continuous part spanned by Eisensteinseries as discussed in sect41 The discrete part of the spectral decomposition of L2(ΓH) isspanned by Maaszlig cusp forms as discussed in sect42 these are the eigenfunctions that arethe primary object of study in arithmetic quantum unique ergodicity which is now a theo-rem thanks to work of Lindenstrauss [38] and Soundararajan [61] Those Maaszlig cusp formsthat are also eigenfunctions of theHecke operators on themodular surface are real-analyticbut not holomorphic and are the central automorphic objects on GL(2) but remain con-siderably more mysterious than their holomorphic counterparts Even writing down evenHecke-Maaszlig forms for ΓH has proven elusive and associating a Galois representation to aMaaszlig form is not known in general

Additionally any connection to motivic cohomology natural in the case of holomorphicHecke eigencuspforms is absent In particular this means that Delignersquos proof [11] of theRamanujan conjecture on Fourier coefficients (ie Hecke eigenvalues) of holomorphic cuspforms does not apply to Maaszlig cusp form coefficients Indeed though the Ramanujan con-jecture for a Hecke-Maaszlig cusp form f would follow from the automorphy of all symmetricpower L-functions L

(s symr f

) this is likely out of reach in the near future and the best

bounds towards GL(2) Ramanujan come from harmonic analysis and known automorphyof symmetric power L-functions This is discussed further in sect43

4 OWEN BARRETT

Nevertheless it is possible to formulate a trace formula for L2(ΓH) using methods notunlike in the case of S a compact hyperbolic surface The Selberg trace formula in this caseconnects spectral data of automorphic forms in L2(ΓH) to the geometry of themodular sur-face This duality can be seen as complementary in the cases where it is relevent to compareto the Grothendieck-Lefschetz fixed-point theorem which connects geometric fixed pointsto motivic cohomology There is another trace formula the Kuznetzov trace formula thathas been developed very explicitly on GL(2) and more recently on GL(3) and which is afavorite tool of analytic number theorists when dealing with Maaszlig forms and spectral auto-morphic L-functions since it expresses a trace of spectral data in terms of familiar charactersums and special functions

This article seeks to expose and elucidate some of the spectral questions surrounding au-tomorphic forms on GL(2) and provide a conceptual introduction to the Selberg trace for-mula in this setting from a classical perspective It should be viewed as complementary to atechnical discussion of the trace formula as a complete technical discussion is not the goalhere Instead we choose to focus on (a) introducing themany arithmetic applications of thetrace formula and spectral theory and (b) providing a tour of key problems in this area

In sect2 we introduce the trace formula in its original setting of a Riemannian manifoldThis is technically more straightforward and prepares us to tackle the trace formula in thenoncompact arithmetic case in sect3 We then proceed to describe some of the properties andmysteries of the spectrum of the Laplacian on the modular surface in sect4

Before all of this however we pause to recall the Poisson summation formula and itsproof which provided the basic concept that Selberg initially sought to generalize We askthat f isin C2(R) and either f f prime f primeprime isin L1(R) or | f (x)| ≪ 1(1 + |x|)1+δ We could insteadrequire that f f isin L1(R) and have bounded variation where f is the Fourier transformwhich is given by

f(ξ)= int

Rf (x)e(ndashxξ) dx

(Throughout e(z) = e2πiz) Then the Poisson summation formula is

summisinZ

f (m) = sumnisinZ

f (n)

The Poisson summation formula is of constant use in number theory since it allows one topass back and forth between the time and frequency domains depending on which is easierto understand Itmay be proved in a straighforwardway by expanding the periodic function

g(y) = summisinZ

f (y + m)

as its Fourier series and setting y = 0 However to draw a better analogy with the Selbergtrace formula we give the proof of Lapid [35]


We first form the convolution operator R( f ) on L2(T) (T = ZR) For a φ isin L2(T)

R( f )φ(x) = intR

f (y)φ(x + y) dy = intR

f (y ndash x)φ(x) dy

= intTsum

nisinZf (y + n ndash x)φ(y) dy = int

TK f (x y)φ(y) dy

where K f (x y) = sumnisinZ f (y + n ndash x) isin Cinfin(T2)We now compute the trace of R( f ) in two different ways First

tr R( f ) = intTK f (x x) dx = sum

nisinZf (n)

On the other hand R( f ) may be diagonalized using the orthonormal basis en(z) = e(nz) Itis immediately checked that R( f )en = f (n)en This shows that

tr R( f ) = sumnisinZ

f (n)

and we have arrived at the Poisson formulaThe basic idea of computing the trace of an appropriately-chosen integral operator in two

different ways shall prove quite fruitful in the development of more sophisticated versionsof the lsquotrace formularsquo We conclude the introduction with what is probably the most famousapplication of the Poisson summation formula namely the proof of the functional equationof ζ(s) using the modularity of Jacobirsquos theta function

θ(z) = sumnisinZ

endashπn2z real z gt 0

If we choose f (x) = endashπx2 in the Poisson summation formula then since f (x) = f (x) wehave that for y gt 0

sumnisinZ

endashπn2x =1radicy sumnisinZ

endashπn2y

It is then straightforward to obtain the identity

θ(1z) =radic

z θ(z)Riemann expressed ξ(s) as the Mellin integral

2πndashs2Γ(s2)ζ(s) = 2ξ(s) =infin

int0(θ(u) ndash 1)us2 du

u real s gt 0

Breaking up the integral and changing variables we arrive at

ξ(s) +1s+

11 ndash s

=12

infin

int1(θ(u) ndash 1)(us2 + u(1ndashs)2)

duu

6 OWEN BARRETT

Since the right-hand side is invariant under s harr 1 ndash s so too is the left-hand side Bothare analytic by the exponential decay of θ(u) (cf [14] for more details) This concludes ourdalliance with theta functions and our review of Poisson summation

2 The Trace Formula Compact Case

In his epochal 1956 paper [56] Selberg announced a lsquogeneral relation which can be con-sidered as a generalization of the so-called Poisson summation formularsquo This lsquogeneral rela-tionrsquo which he referred to as a the lsquotrace formularsquo began as a way to study the spectrum ofdifferential operators of finite order on a Riemannian space S that are invariant by a locallycompact group of isometries of S When S = Σg is a compact hyperbolic surface of genusge 2 Σg may be formed as a quotient π1(Σg)H where π1(Σg) sub PSL(2R) the fundamentalgroup of Σg (and group of deck transformations with respect to the universal cover H) is astrictly hyperbolic Fuchsian group For ease of exposition and to conform with [22 Chap-ter 1] whose techniques we refer to throughout we restrict to this case for the remainder ofthis section

We recall that H is a Riemannian manifold with the Poincareacute metric ds2 = dx2+dy2y2 and

volume form dμ(z) = dx dyy2 The Poincareacute metric has constant negative Gaussian curvature

K = ndash1 The Laplacian onH is the familiar operator

Δ = y2(

partsup2partsup2x

+partsup2partsup2y

)

It is the unique fundamental differential operator invariant under the action of SL(2R) (ieall others are polynomials in Δ) Since it is invariant by SL(2R) and hence by the Fuchsiangroup π1(Σg) Δ descends to the operator D on a Riemann surface Σg via the projectionH minusrarr Σg Explicitly if ξ and η are local coordinates onΣg such that ds2 = a dξ2+2b dξ dη+c dη2 then D is given explicitly by

(21) D =1

ac ndash b2

[partpartξ

(cpartpartξ ndash bpartpartηradic

ac ndash b2

)+

partpartη

(apartpartη ndash bpartpartξradic

ac ndash b2

)]

Since Σg is compact(a) the spectrum of D is discrete

0 = μ0 lt μ1 leμ2 le middot middot middot limnrarrinfin

μn = infin

(b) corresponding to the eigenvalues μn we can construct an orthonormal basisφn

for L2(Σg) satisfying Dφn + μnφn = 0(c) the normalized eigenfunctions φn are real-valued(d) we have a version of Besselrsquos inequality

infinsumk=1

1μ2k

lt infin


It follows fromessentially the classicalHilbert-Schmidt theorem that theFourier seriessuminfinn=0 cnφn

of any C2(Σg) function converges uniformly and absolutely cf [15 1 p383] and [27 1p234ndash235] We renotate as λn = ndashμn to conform to Selbergrsquos [56] notation Then

φn isin Cinfin(π1(Σg)H) and Dφn = λnφn

As a consequence of the Stone-Weierstrass theorem as in [26 p151] we have the spectraldecomposition

(22) L2(π1(Σg)H) =infinoplus

n=0

[φn]

It was Selbergrsquos crucial insight that the spectral theory of D the Laplacian on Σg can beformulated in terms of integral operators of the form

(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)

where dμ(w) denotes the invariant element of volume derived from themetric and the ker-nel is necessarily π1(Σg)-invariant ie

k(zw) = k(w z) andk(mzmw) = k(zw) for w z isin Σg m isin π1(Σg)

Such a k is called a lsquopoint-pair invariantrsquo The purpose of the rest of this section is to makeexplicit the way in which the operator R( f ) provides a window into the spectrum of theLaplacian on a compact hyperbolic surface We have selected this notation for R( f ) to be inclose analogy to the example of Poisson summation given in the introduction Proofs maybe found in [22 Chapter 1]

We begin by recalling that Σg admits a Fuchsian model as a quotient of H by π1(Σg) aFuchsian group (discrete subgroup of PSL(2R)) The quotient space π1(Σg)H has a poly-gon F as a respresentative element that is known as the standard fundamental polygon with4g sides where g is the genus of Σg Additionally

area(Σg) = μ(F) = 4π(g ndash 1)

with μ as aboveNext given a Φ isin C00(R) there is an associated point-pair invariant k(zw) given by

(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)

Proposition 21 ([22 Ch 1 Prop 31]) Let f be any eigenfunction ofΔ onHwith eigenvalue λThen

intH

k(zw) f (w) dμ(w) = Λ(λ) f (z)

where Λ(λ) depends solely on λ and the test function Φ

8 OWEN BARRETT

If we form the automorphic kernel function

K(zw) = sumσisinπ1(Σg)

k(σzw) zw isin H

then provided as always Φ is a real test function with complex support K(zw) is actuallya sum of uniformly bounded length K(zw) = K(w z) and K(zw) is π1(Σg) times π1(Σg)-invariant

Returning now to the integral operator R( f ) as in (23) and with Φ isin C00(R) as usualthen it can be shown that R( f ) is a bounded linear operator on L2(π1(Σg)H) and may beexpressed as

R( f )(z) = intFK(zw) f (w) dμ(w) for any f isin L2(π1(Σg)H)

It follows (see [70 p277]) thatR( f ) is anoperator ofHilbert-Schmidt typeonL2(π1(Σg)H)This allows us to state a few facts about R( f ) that justify our choice of R( f ) and make thenotion of a lsquotrace formularsquo both viable and deeply significant

Proposition 22 ([22 Ch 1 Prop 38]) Let Φ isin C200(R) Then

(a) K(zw) = suminfinn=0 Λ

(λn)φn(z)φn(w) with uniform absolute convergence onHtimes H

(b) suminfinn=0∣∣Λ (λn

)∣∣ = infin(c) intFK(z z) dμ(z) = suminfin

n=0 Λ(λn)

Proof In view of the spectral decomposition (22) a consequence of the Fourier expansionof C2 functions on Σg times Σg and the identity

intFK(zw)φn(z) dμ(z) = int

Σg

k(zw)φn(z) dμ(z) = Λ(λn)φn(w)

Thesignificance of Proposition 22 is immediately illustratedwhen considered alongside thespectral decomposition (22) and Proposition 21 which allow us to conclude that

tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

where K(z z) = sumσisinπ1(Σg) k(σz z) as above The Selberg trace formula arises from a term-by-term expansion of the last integral This expansion is somewhat technical and full detailsmay be found in [22 sect14ndash16] We limit our concern to providing some flavor for the ex-pansion and characterizing its general shape without getting bogged down in technicalities

We notate conjugacy classes in π1(Σg) as [σ] We observe that for a fixed σ isin π1(Σg)gndash1σg = hndash1σh hArr hgndash1σghndash1 = σ hArr ghndash1 isin Z(σ) hArr g isin Z(σ)h where Z(σ) is the


centralizer of σ isin π1(Σg) We may then write

tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct

sumρisinZ(σ)π1(Σg)

intFk(ρndash1σρz z) dμ(z)

= sum[σ]


intFk(σρz ρz) dμ(z)

= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)

It can be shown thatcupρisinZ(σ)π1(Σg)

ρ(F)

is a fundamental region for the Fuchsian group Z(σ) and that the integral of k(σz z) overany reasonable fundamental region for Z(σ) is independent of the choice of fundamentalregion We are then justified in letting F[ZΓ(σ)] denote any fundamental region for Z(σ)and writing

tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)

This sum is absolutely convergent since as remarked above K(zw) is actually a sum ofuniformly bounded length It becomes clear that to compute the other side of the traceformula we must select a fundamental region F[ZΓ(σ)] in such a way that we might be ableto execute the resulting computations effectively

For an arbitrary element σ isin π1(Σg) sub PSL(2R) σ is either hyperbolic or σ = I Thisillustrates how the case under consideration is almost a lsquotoy casersquo in any event it is highlynon-generic in the sense that in general there will be a large variety of other kinds of con-tributions to these orbital integrals than just the ones we will see here Also since we areworking on H there is only one fundamental invariant operator a further simplification

10 OWEN BARRETT

This is perhaps why Dennis Hejhal said lsquoOf course we wonrsquot really understand the traceformula until it is written down for SL(4 Z)rsquo

If σ is hyperbolic then Z(σ) is a cyclic subgroup of π1(Σg) with a generator σ0 that isuniquely determined by σ Eventually putting z = ηw with η isin PSL(2R) we may write

E = intF[ZΓ(σ)]

k(σz z) dμ(z) = intF[ZΓ(σ)]

k(σz z) dμ(z)

= intηndash1F[ZΓ(σ)]

k(σηw ηw) dμ(w)

= intF[ZΓ(ηndash1ση)]

k(ηndash1σηww) dμ(w)

We may choose η isin PSL(2R) such that

ηndash1σ0η(w) = N(σ0)w 1 lt N

(σ0)lt infin

Then

ηndash1ση(w) = N(σ)w

where N(σ) is the multiplier or characteristic constant of the transformation σ It can beshown that a fundamental region for ηndash1σ0η is given explicitly by

F[ZΓ(ηndash1σ0η)] =1leimagew lt N(σ0)

This choice of fundamental region allows us to compute this orbital integral fully explicitlyand we obtain

Proposition 23 ([22 Ch 1 Prop 63]) For hyperbolic σ isin π1(Σg) and denoting the generatorof the cyclic group Z(T) by σ0 we have

intF[ZΓ(σ)]

k(σz z) dμ(z) =logN(σ0)

N(σ)12 ndash N(σ)ndash12g(logN(σ))

where

g(u) =infin

intx

Φ(t)radict ndash x

dt for x = eu + endashu ndash 2 and u isin R

The other case is when σ = I In this case F[ZΓ(I)] = F[ZΓ(π1(Σg))] = F and

intFk(z z) dμ(z) = Φ(0)μ(F)

It is quite technical to get ahold of Φ(0) but with perseverence one finds that


Proposition 24 ([22 Ch 1 Prop 64]) If Φ isin C200(R)

Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr

This is enough to arrive at a preliminary version of the Selberg trace formula for a com-pact hyperbolic surface Σg However it requires some more work to arrive in the final formNamely to arrive at Selbergrsquos version of the trace formula [56 p74] for Σg we need to com-pute the abcissa of convergence of a particular spectral sum ie we would need to showthat

inf

ν ge 1

infinsumn=1

∣∣λn∣∣ndashν lt infin

= 1

This plus an approximation argument yields the final form of the Selberg trace formulawhich is considerablymoreflexible than theoneobtained so far Inparticular we can replacethe as-of-yet-implicitly-defined function Λ with a class of even test functions satisfying thefollowing standard conditions

(1) h(r) is analytic on∣∣image r∣∣ le 1

2 + δ(2) h(ndashr) = h(r) and(3) |h(r)| ≪

(1 +∣∣real r∣∣)ndash2ndashδ for some δ gt 0

We recall some notation and state the final version of the trace formula straightaway TheL2(π1(Σg)H) spectrum of Δ is

λn where

0 = λ0 gt λ1 ge λ2 ge middot middot middot limnrarrinfin

λn = ndashinfin

Theorem 25 (The Selberg trace formula for a compact hyperbolic surface [22 Ch 1 The-orem 75]) Let Σg be a compact hyperbolic surface let Φ isin C2

00(R) and suppose that h(r) satisfiesthe above assumptions Then writing λn = 1

2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]

logN(σ0)N(σ)12 ndash N(σ)ndash12

g(logN(σ))

where

g(u) =12π

infin

intndashinfin

h(r)endashiru dr u isin R

The final form of the trace formula given in Theorem 25 is tantalizing when compared tothe explicit formula attached to the zeta function or any L-function If γ isin C is an ordinate

12 OWEN BARRETT

of a nontrivial zero of ζ(s) ie ζ(ρ) = 0 and ρ = 12 + iγ then the explicit formula takes the

form of the following sum over ordinates of critical zeros for some test function h

sumγ

h(γ) = h(12)ndash g(0) log π +

12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n

where Λ is von Mangoldtrsquos function and g is related to h as above The Riemann hypothesisis equivalent to the positivity of the right hand side of the explicit formula for convolution-type functions g(u) of the form

g(u) =infin

intndashinfin

g0(u + t)g0(t) dt

The hope then would be to gain some insight into ζ(s) and the Riemann hypothesis bystudying trace formulaeligattached to arithmetic groupsΓ such as the arithmetic latticePSL(2Z)This is sometimes referred to as lsquoHilbert and Poacutelyarsquos dreamrsquo in any case this resemblencemotivates the study in the next section

3 The Trace Formula Arithmetic Case

Wenowarrive at the arithmetic case that is theprimary concernof ourdiscussion namelythe noncompactmanifold ΓH where Γ = PSL(2Z) is the full modular group Arthur [3 p2ndash3] presents eloquent motivation for the general study of Laplace-Beltrami operators at-tached to reductive algebraic groups He explains that the eigenforms of these differentialoperators automorphic forms are expected to lsquocharacterize some of the deepest objects ofarithmeticrsquo The case of ΓH is the most classical and home to so much arithmetic As be-fore we will consider this space in as elementary and explicit language as possible in thespirit of Hejhal whose development we shall follow In [23] Hejhal formulates three ver-sions of the Selberg trace formula for noncompact quotients of the half-plane by Fuchsiangroups whose fundamental regions have finite non-Euclidean area Recall that a Fuchsiangroup acts on H by fractional linear transformations Among these examples we have thecase of the full modular group and congruence subgroups such as the principal congruencesubgroup of level N

Γ(N) =(

a bc d

) a d equiv plusmn1 mod N and b c equiv 0 mod N

the congruence subgroup

Γ0(N) =(

a bc d

) c equiv 0 mod N


and the intermediate congruence subgroup

Γ1(N) =(

a bc d

) a equiv d equiv 1 mod N and c equiv 0 mod N

To these congruence subgroups there are associatedmodular curvesX(N) = Γ(N)H X0(N) =Γ0(N)H and X1(N) = Γ1(N)H There is a formula for the number of cusps of variousmodular curves for example∣∣cusps of Γ0(N)

∣∣ = sumd|N

φ(gcd(d Nd))

where φ is Eulerrsquos totient functionLet G denote any Fuchsian groupwhose fundamental region F has finite (non-Euclidean)

area We take a moment to review some vocabulary A matrix A isin SL(2C) is said to be(i) loxodromic when tr A isin [ndash2 2](ii) hyperbolic when tr2 A isin (4infin)(iii) elliptic when tr A isin (ndash2 2) and(iv) parabolic when tr A = plusmn2 and tr A isin I ndashI

When developing a theory of group actions on hyperbolic space we think of matrices inPSL(2R) as fractional linear transformations of the half-plane H We recall the notion ofnormal form for a hyperbolic transformation L isin PSL(2R) which has two fixed points sayξ and η Setting

g(z) =z ndash ξz ndash η

then the transformation gLgndash1 fixes 0 and infin and is thus a dilation κz This complex numberκ is called the multiplier of L and satisfies

λ12 + λndash12 =∣∣tr L∣∣

To be consistent with [22 23] we use the notation N(L) = λ for the multiplier of a hyper-bolic transformation L

Let χ be an r times r unitary representation of G We then define the L2 space

L2(GH χ) =

f f is measurable onH

f (σx) = χ(σ) f (x) for σ isin G

intF f (z) t f (z) dμ(z) lt infin

(Here we regard the functions f as columnvectors) It follows fromthe assumption area(GH) ltinfin that G is a finitely-generated Fuchsian group of the first kind We say that a point a isin His an elliptic fixpoint for G if there is an elliptic element σ isin G such that σ stabilizes a Theattached isotropy subgroup Ga then reduces to a finite cyclic group ie

Ga =σ isin G σ(a) = a

14 OWEN BARRETT

is generated by an elliptic element of G of finite orderWe call a point b isin R cup infin a cusp if there is a parabolic transformation η isin G such that

η(b) = b In this case the isotropy subgroup Gb is infinite cyclic with parabolic generatorTwo elliptic fixpoints or cusps are deemed equivalent when one can be taken to the other byan element of G

The above assumption on the finiteness of the area of a fundamental region for G impliesthat the number of inequivalent elliptic fixpoints and the number of inequivalent cusps areboth finite We denote these integers s and κ respectively

The analogue of the hyperbolic Laplacian for automorphic forms of weight k is

Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx

see Maaszlig [44] Among other things recall that an automorphic form of weight k on sayΓH is an eigenfunction of Δk We will be somewhat relaxed in this exposition about ournotation for the Laplace-Beltrami operator on ΓH It descends as in the compact case (21)but we will just persist in notating it as Δ for sake of notational simplicity

On theway towardsdeveloping thefinal versionof Selbergrsquos classical trace formulaHejhal [23]formulates progressively more general versions Explicitly with r the dimension of the uni-tary representation χ of G s the number of inequivalent elliptic fixpoints κ the number ofinequivalent cusps and k the weight which specifies the differential operator Δk Hejhalrsquosthree formulations of the trace formula on L2(GH χ) for G a Fuchsian group of the firstkind with area(GH) lt infin are given by the letters A B and C corresponding to the follow-ing choice of parameters

κ s r kA 1 finite 1 0B finite finite finite 0C finite finite finite arbitrary real

Table 1 Versions of the Selberg trace formula

We note that the modular surface PSL(2Z)H has a cusp at infin and elliptic fixpoints 1and 0 hence κ = 1 and s = 2 Version A of the trace formula contains the central ideasin the simplest form and since it addresses the case of the modular surface we will presentonly this case The technicalities required to arrive at versions B and C are substantial andmay be found in [23 Ch 8 9] respectively The case of a Riemann surface discussed in sect2corresponds to κ = 0 s = 0 χ trivial and k = 0

31 Introducing the SelbergTrace Formula onΓH In this section Γ denotes any Fuchsiangroup whose fundamental region has finite area and one cusp but we might as well justthink about it as PSL(2Z) Without loss of generality we can place the single cusp at infinthen from the above the stabilizer Γinfin of the cusp is infinite cyclic From the action of Γ byfractional linear transformations we see that Γinfin is generated by the matrix S =

( 1 10 1)


Thevalue that the character χ assumes on the generator S changes thematter of formulat-ing the Selberg trace formula for ΓHdrastically For while the spectral theory of L2(ΓH χ)is not far away from the case of a compact surface when χ(S) = 1 it is quite different whenχ(S) = 1 and the usual attempt to characterize the spectrum in terms of integral operatorsfails badly It is illustrative to first discuss howwewould like to formulate the spectral theoryof L2(ΓH χ) in general this approach actually works when χ(S) = 1 The basic idea is totry to apply the resolvent formalism to the spectral theory of Δ on ΓH This can be under-stood as a classical exercise in applying Fredholm integral operator techniques to a problemin PDE

As in the compact case we would like to reformulate the spectral theory of L2(ΓH χ) interms of integral operators If this is possible thenwe should be able to useGreenrsquos functionsas resolvents to characterize the spectrum in a standard way More concretely here are thesteps we would like to take to understand the spectral theory of L2(ΓH)

(a) Construct a Greenrsquos function for the PDE Δu + s(1 ndash s)u = 0 on L2(ΓH χ) that willserve as the integral kernel of a resolvent for the Laplace-Beltrami operator on ΓHIn particular the eigenfunctions of the operators arising from the Greenrsquos functionscoincide with those of Δ

(b) Show that the corresponding integral operators have a discrete countable spectrum(c) Verify that the conditions of the Hilbert-Schmidt theorem are satisfied for the inte-

gral operators with Greenrsquos functions as kernels(d) Conclude that the spectrum of Δ on ΓH is discrete and spanned by a countable

orthonormal basis of eigenfunctions(e) Form a new automorphic kernel function K in analogy with sect 2 chosen so that the

integral of the trace on the one hand coincides with the trace of the spectrum ofΔ enveloped by a test function while on the other hand can be expressed in termsof orbital integrals of a point-pair invariant that are tractable and can be computedfairly explicitly

(f ) Explicitly compute the resulting orbital integrals

Fortunately this goes through in the case that χ(S) = 1 It absolutely does not in the casethat χ(S) = 1 which is the case of most classical interest to number theorists It is easyto see that it doesnrsquot work out harder to understand why To see that it doesnrsquot work outassume all steps prior to (e) and following [23 Ch 6 sect9] consider the standard point-pairinvariant k(zw) as in 24 where Φ isin C00(R+) Φ(0) = 1 and Φ(t) ge 0 In obvious analogywith before form the kernel function

K(zw χ) = sumσisinΓ

χ(σ)k(z σw)

The compact support of Φ(t) ensures convergence of K(zw χ) Repairing to [23 Ch 6Prop 66] shows thatK(zw χ) equiv 0near z = iinfinforallw isin H hence thatK(zw χ) isin L2(ΓH χ)

16 OWEN BARRETT

If the spectrumof L2(zw χ) were discrete and reasonable wewould then have an orthonor-mal decomposition

L2(ΓH χ) =infinoplus

n=0

[φn]

for eigenfunctions φn This spectral decomposition immediately implies the spectral expan-sion

K(zw χ) =infinsumn=0

h(rn)φn(z)φn(w) forallw isin H

In analogy with sect 2 we will most definitely need sum∣∣h (rn)∣∣ lt infin to develop a sensible notion

of trace The hope would be that by takingΦ sufficiently smooth we would be able to attainthis (necessary) condition in which case we could make a statement like

intFK(z z χ) dμ(z) =

infinsumn=0

h(rn)

But the integral over the fundamental region on the left hand side diverges To see this wefind some α such that Φ(t) ge 1

2 for t isin [0 α] Then

K(z z χ) = sumσisin[S]

χ(σ)k(z σz) + sumσ isin[S]

χ(σ)k(z σz) for z isin F

Taking y = image z sufficiently large and assuming that for y large F begins to look like therectangular region [0 1] times [y infin] it can be seen that due to the construction of k(w z) thesum over σ isin [S] does not contribute due to the compact support of Φ We are left with thesum over the stabilizer of the cusp which for large values of y

K(z z χ) =infinsum

n=ndashinfink(z Snz) =

infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy

y2= +infin for some Y

The convergence issues with our naiumlve effort indicate that our assumptions of a discretewell-behaved spectrum when χ(S) = 1 has gotten us in trouble We will re-assume this mat-ter in the next section but the above provides evidence for a continuous spectrum Character-izing this continuous spectrum decomposing it into generators and arriving at a spectraldecomposition of L2(ΓH) will be the primary objective of our efforts to address the caseχ(S) = 1 in what follows Let

(31) δ(χ) =

1 χ equiv 10 χ equiv 1


We will conclude by stating the spectral decomposition due to Selberg

(32) L2(ΓH χ) = δ(χ)Coplus L2cusp(ΓH) oplus L2

(ΓH) oplus L2cont(ΓH)

In order δ(χ)C are the constant functions present only when χ equiv 1 L2cusp(ΓH) is the

space spanned by cusp forms as in the compact case or the case when χ(S) = 1 L2(ΓH) is

spanned by residues of GL(2) Eisenstein series It has another characterization that we willbe useful later Finally L2

cont(ΓH) is the continuous spectrum Wewill give amore involveddescription in the next section

As a final remark we return to the case that is foremost in our minds that is the case ofΓ = SL(2Z) with trivial character Then the spectral decomposition is simply

L2(SL(2Z)H) = Coplus L2cusp(SL(2Z)H) oplus L2

cont(SL(2Z)H)

We note that the contribution to the discrete spectrum from residues of Eisenstein series isnot present This is because the only pole of the Eisenstein series in real s ge 1

2 is at s = 1 andthe residue there is a constant function Therefore it doesnrsquot contribute anything new

32 The Spectral Decomposition of ΓH In this section we present the key techniquesused to decompose the (discrete) spectrum in the case χ = 1 and we elaborate on how toarrive at the decomposition (32) The references throughout are [23 Ch 6ndash7] and [15 Ch10]

First we address the easier case of χ(S) = 1 The goal is to use Hilbert-Schmidt to char-acterize the spectrum of L2(ΓH χ) To do this we must transfer the spectral theory of Δ tothat of some easier-to-understand integral operators That is we would like to introduce aGreenrsquos function for the PDEΔu+ s(1ndash s)u = 0 on L2(ΓH χ)1 There are a raft of conditionswe would like such a function ks(z z0) to satisfy Referring to [23 Ch 6 sect6] we insist that

(a) ks(z z0) be a point-pair invariant(b) ks(z z0) should be Cinfin for z isin H ndash z0(c) ks(z z0) = 1

2π log∣∣z ndash z0

∣∣ + O(1) near z = z0(d) Δks(z z0) + s(1 ndash s)ks(z z0) = 0 for z isin H ndash z0(e) ks(z z0) should be as small as possible when z rarr partH

The reason for the last condition is that Δ is singular along partH whichwill introduce difficul-ties Using the theory of PDE and Eulerrsquos hypergeometric differential equation in particularwe can prove the following statement about the Fourier coefficients of an eigenfunction ofΔ onH

Proposition 31 ([23 Ch 6 Lemma 42 amp Prop 43]) Suppose that z0 isin H 0le r1 lt r2 le1s isin C ndash 0 ndash1 ndash2 ndash3 Assume f is C2 and satisfies Δf + s(1 ndash s)f = 0 on the set

N(z0 r1 r2 =z isin H r1 lt ∣w∣ lt r2

where w = reiθ =

z ndash z0z ndash z0

1The introduction of the extra s(1 ndash s)u term is a standard trick for attacking eigenvalues for which theGreenrsquos functions may not otherwise converge cf [22 note 4 p354])

18 OWEN BARRETT

Define

fn(ξ) = ξn(1 ndash ξ2)sF(s + n s 1 + n ξ2)

Γ(1 + n)for |ξ| lt 1 and

gn(ξ) = ξn(1 ndash ξ2)sF(s + n s 2s 1 ndash ξ2)

Γ(2s)for0 lt |ξ| lt 1

∣∣Arg ξ∣∣ lt π

4

where F(a b c x) is the hypergeometric function a solution of Eulerrsquos differential equation

x(1 ndash x)uprimeprime +[c ndash (a + b + 1)x

]uprime ndash abu = 0

Then(i) f (z) = suminfin

n=ndashinfin cneinθ for r1 lt r lt r2(ii) cn(r) satisfies the differential equation

(33) uprimeprime + 1ruprime +

(4s(1 ndash s)(1 ndash r2)2

ndashn2

r2

)u = 0 on r1 lt r lt r2

while fn(r) and gn(r) satisfy (33) on 0 lt r lt 1(iii) cn(r) = An f |n|(r) + Bn g|n|(r) for appropriate constants An and Bn

Using Proposition 31 we may conclude that ks(z z0) must be a linear combination of f0(r)and g0(r) To satisfy the growth condition (e) in the list of conditions above and by examin-ing the behavior of g0(r) at r = 0 using differential equations techniques and examining thebehavior of the linearly independent solutions f0 and g0 to (33) we find that it behooves usto set

ks(z z0) = ndashΓ(s)2

4πg0(r)

since for real s gt 12 the ODE (33) has a regular singular point at r = 1 and setting the

coefficient of f0 in the linear combination constituting ks(z z0) to zero gives us the smallestpossible growth near partH (see [23 p30] for more details) That is we set

Definition 32 ([23 Ch 6 Def 61]) Forreal s gt 12 let

ks(zw) = ndashΓ(s)2

4πΓ(2s)

(1 ndash∣∣∣z ndash wz ndash w

∣∣∣2)sF(

s s 2s 1 ndash∣∣∣z ndash wz ndash w

∣∣∣2)

With this definition ks(zw) is the free-space Greenrsquos function for Δu+ s(1 ndash s)u = 0 Naturallywe would now like to lsquoprojectrsquo ks(zw) down to a function on the coset space ΓH To con-struct the Greenrsquos function on L2(ΓH) we would like to do the most naiumlve thing which isto define the Poincareacute series

Gs(zw χ) = sumσisinΓ

χ(σ)ndash1ks(σzw)


Fortunately this is not at all far off from what we end up doing We would like to insistthat Gs(zw χ) be absolutely convergent since a well-behaved definition of a Poincareacute se-ries on ΓH obviously shouldnrsquot depend on the order of summation We have the followingconvergence result

Proposition 33 ([23 Ch 6 Prop 62 amp 63]) Up to dropping a finite number of terms in theseries sumσisinΓ χ(σ)ndash1ks(σzw) converges uniformly and absolutely on compact subsets ofHtimesHtimes s isinC real s gt 1 On the other hand the infinite series

sumσisinΓ

∣∣ks(σzw)∣∣

is divergent whenever z equiv w mod Γ andreal s = 1

Motivated by Proposition 33 we proceed to define the automorphic Greenrsquos function onΓH by

Definition 34 (The automorphic Greenrsquos function [23 Ch 6 Def 64])


χ(σndash1)ks(σsw) for (zw) isin Htimes H andreal s gt 1

We usually assume z equiv w mod Γ for safety Next we summarize some nice properties ofGs(zw χ)

Proposition 35 ([23 Ch 6 Prop 65]) Assume thatreal s gt 1 Then(a) Gs(zw χ) = Gs(w z χ) while Gs(zw χ) = Gs(w z χ)(b) Gs(σz ηw χ) = χ(σ)Gs(zw χ)χ(ηndash1) for σ η isin Γ(c) Gs(zw χ) is Cinfin in both variables when z equiv w mod Γ(d) Gs(z z0 χ) satisfies Δu + s(1 ndash s)u = 0 on the set z = z0 mod Γ(e) when Γz0 = I we have

Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0

(f ) when Γz0 is infinite cyclic generated by an elliptic R isin Γ of order ν we have

Gs(z z0 χ) =

0 χ(R) = 1ν2π log

∣∣z ndash z0∣∣ + O(1) near z = z0 χ(R) = 1

The significant points of Proposition 35 is that Gs(z z0 χ) has at most a logarithmic sin-gularity near the diagonal and is an eigenfunction of the Laplacian on ΓH The point isthat not unlike in the case of a compact Riemann surface the spectral theory of Δ can bereformulated in terms of integral equations involving a kernel with at most a logarithmicsingularity The next proposition shows that Gs(zw χ) must be as well-behaved near thecusp as we could want

20 OWEN BARRETT

Proposition 36 ([23 Ch 6 Prop 66]) Let EtimesK be a compact subset ofHtimes real s gt 1 Assumethat 0lereal zle1 and w isin E s isin K Then

sumσisinΓ

∣∣ks(σzw)∣∣ zrarriinfinminusminusminusminusrarr 0 unformly

The next task is to obtain a Fourier theory for the automorphic Greenrsquos function This isextremely technical and relies heavily on the theory of Bessel functions which can also beused to effect a kind of lsquoseparation of variablesrsquo in the same way as was achieved in Propo-sition 31 using hypergeometric functions Once this separation of variables is achieved wecan express a Fourier expansion for Gs(zw χ) in terms of I- and K-Bessel functions Thefull calculation is performed in [23 Ch 9 sect4 6] In keeping with the goal of emphasizingconcept we do not reproduce pages of integrals over cosets and state the main result

Proposition37 (Explicit expression for automorphicGreenrsquos functions [23 p 41ndash42]) Letα isin [0 1) be defined implicitly by χ(S) = e(α) Set μ = minn+α =0 |n + α| Then

Gs(zw χ) =δ0αF0(w s χ) y1ndashs

1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)

=δ0αF0(w s χ) y1ndashs

1 ndash 2s+ O

(endash2πμy) where

Fn(w s χ) = sumW0isinΓinfinΓ

χ(Wndash1

0)image(W0w)

12 Isndash 1

2

(2π|n + α|image(W0w)

)times e((n + α)real(W0w)) when n + α = 0 and

F0(w s χ) = sumW0isinΓinfinΓ

χ(Wndash10 )image(W0w)s when n + α = 0

Our attention turns next to the functions Fn(z s χ) Let R = s isin C real s gt 1 ThenFn(z s χ) is defined on H times R using Proposition 37 We summarize some facts about thebehavior of these functions on compact subsets of R

Proposition 38 ([23 Ch 6 Prop 81]) Let K sub R be compact Then(a) Fn(z s χ is not affected by ambiguities in W0(b) Fn(z s χ) converges uniformly amp absolutely onHtimes R compacta(c) Fn(σz s χ) = χ(σ)Fn(z s χ) for σ isin Γ(d) Fn(z s χ) is a Cinfin solution of Δu + s(1 ndash s)u = 0 onH(e) the lsquotruncatedrsquo series

Fn(z s χ) = Fn(z s χ) ndash

y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys


tends uniformly to 0 wheneverimage z rarr +infin and s isin K

321 χ(S) = 1 We are now in an optimal position to move forward with the spectral de-composition of L2(ΓH) in the case that χ(S) = 1 We begin by noting that crucial foran application of Hilbert-Schmidt the automorphic Greenrsquos function is square-integrableThroughout this section χ(S) = 1 is assumedProposition 39 ([23 Ch 6 sect10]) Let K sub R Then

(34) intF

∣∣Gs(zw χ)∣∣2 dμ(z)

is uniformly bounded for s isin K and w isin H (The bound depends on χ K and Γ only) Addition-ally (34) is continuous in w and tends to 0 wheneverimagew rarr infin

Next we have the crucial proposition that eigenfunctions of the Fredholm integral operator

(35) v 7rarr intFGs(zw χ)v(w) dμ(w)

on L2(ΓH χ) are precisely the ones we want

Proposition 310 ([23 Ch 6 Prop 102]) Assume that 1 lt a lt infin Then

v(z) = λ intFGs(zw χ)v(w) dμ(w) with v isin L2(ΓH χ)

if and only ifΔv + a(1 ndash a)v = λv with v isin Cinfin(H) cap L2(ΓH χ)

There is only onemore condition to obtain that the system of eigenfunctions for the integraloperator (35) is complete in C2(H)cap L2(ΓH χ) and thereby arrive at the discrete spectraldecomposition

Proposition 311 ([23 Ch 6 Prop 103]) Suppose that f isin C2(χ)capL2(ΓH χ) and χ(S) = 1Assume ( for simplicity) that f (x + iy) equiv 0 near y = infin Then

f (z) = intFGs(zw χ)h(w) dμ(w)

where h = Δ f + a(1 ndash a)f where 1 lt a lt infin

As we will see Proposition 311 allows us to apply Hilbert-Schmidt theorem since the in-tegral operator with kernel Gs(zw χ) actually is a resolvent for the Laplacian in the aboveway Proposition 310 allows to transfer the characterization we can obtain about our inte-gral operator to a statement aboutΔ Wenowproceed to adiscussionof theHilbert-Schmidttheorem in a somewhatmore basic and restricted setting which nonetheless contains all thekey ideas We state here according to [15 Ch 10 sect4]

22 OWEN BARRETT

First some conditions and notation Suppose K(s t) is a symmetric kernel Suppose weare in the L2 space of a measure space (DΩ σ) where D is a (not-necessarily) compactdomain We use the notation K f to denote the integral convolution

intXK(s t) f (t) dμ(t)

We assume that the kernel K has a weak singularity of the form

K(s t) =κ(s t)rℓ

0le ℓ lt dimD

where κ(s t) is a bounded function and r is the distance between s and t We use the notationK f for the operator

K f = intK(s t)φ(t) dμ(t)

In this notation (35) may be written as Gs(zw χ) v Then we have the classical Hilbert-Schmidt theorem

Theorem 312 (Hilbert-Schimdt) Any function f which can be expressed in the formf = K g

where g is some square integral function and K is a symmetric kernel satisfying the above conditionsmay be expressed as an absolutely and uniformly convergent series

f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)

where the φn are eigenfunctions of K In other words the eigensystem of K forms aHilbert space basisfor L2(D)

In some sense we would like to apply the Hilbert-Schmidt theorem directly to our situa-tion Unfortunately there are additional technicalities relating to the fact that the spectraltheory is a bit different for modular functions on ΓH twisted by the unitary representationχ Fortunately Hejhal [22 Ch 3] formulates the necessary modifications to [15 Ch 10] infull detail and we allow ourselves to conclude that

Proposition 313 (Spectral decomposition of L2(ΓH χ) when χ(S) = 1 [23 p 99]) Theeigensystem of Δ is orthonormal and complete in L2(ΓH χ) ie


n=1

[φn]

(orthonormal and complete)

Δφn + λnφn = 0 and0 lt λ1 le λ2 le λ3 le middot middot middot lim

nrarrinfinλn = infin


Note thatHilbert-Schmidt allows us to conclude asmuch for our integral operator with ker-nel Gs(zw χ) and Propositions 310 allows us to transfer those statements to the operatorwe really care about We emphasize that when χ(S) = 1 the spectrum of Δ is discrete andreasonable

322 χ(S) = 1 It is time to give a more detailed though still mostly non-rigorous andvery incomplete characterization of the spectral decomposition of L2(ΓH χ) in the casethat χ(S) = 1 This case gives us the rich decomposition (32) quite unlike either of theprevious cases The key reference for all statements made in this section is [23 Ch 6 sect9 13amp Ch 7]

We begin by trying to isolate the continuous part of the spectrum We would expect adecomposition like

L2(ΓH χ) = A oplus E

whereA is spannedby theL2 eigenfunctions andE corresponds to the continuous spectrumIt can be shown [23 Ch 10 Claim 91 92] that if φ isin C2(ΓH) cap L2(ΓH χ) is nonzeroand satisfies Δφ + λφ = 0 then λ isin R and in fact λ ge 0 The case λ = 0 occurs iff φ(z) isidentically constant Let 0 = λ0 lt λ1 le λ2 le middot middot middot be the eigenvalues of Δ

∣∣A the restriction

of Δ to A It can also be shown that the eigenfunctions corresponding to the eigenvalues λnare mutually orthogonal and (again using the lsquoseparation of variablesrsquo via Bessel functions)that an eigenfunction φn has the explicit Fourier expansion

(36) φn(x + iy) = bny1ndashs + summ=0

amy12Ksndash 1

2(2π|m|y)e(mx)

and bn = 0 whenever λn ge 14

We let L2(λ) be the subspace generated by those φn satisfying Δφn + λφn = 0 A0(λ) bethe subspace of L2(λ) satisfying the condition b = 0 in (36) and A1(λ) be the orthogonalcomplement of A0(λ) in L2(λ) We then form the spaces

A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)

We arrive at the decompositions

A = δ(χ)Coplus A0 oplus A1

L2(ΓH χ) = δ(χ)Coplus A0 oplus A1 oplus E

We note that the appearance of the δ as defined in (31) is because λ0 is omitted when χ equiv 1Proceeding now to the continuous spectrum since dim

(δ(χ)Coplus A1

)= infin orthogonal-

ity with respect to A1 is the most important condition With ψ(y) isin Cinfin(R+) with compact

24 OWEN BARRETT

support we form the Poincareacute series

θψ(z) = sumW0isinΓinfinΓ

χ(Wndash1

0)ψ(imageW0z

)

θψ(z) is convergent on H compacta and vanishes at the cusp hence θψ(z) isin L2(ΓH χ) Itis not too hard to show to show that θψ perp A0 Namely [23 p73ndash74]

intFθψ(z)φn(z) dμ(z) = sum

W0

χ(Wndash10 ) int

Fφ(imageW0z)φn(z) dμ(z)

= sumW0intFφ(imageW0z)φn(W0z) dμ(z)

= sumW0

intW0(F)

ψ(image ξ)φn(ξ) dμ(ξ)

=infin

int0

1

int0ψ(y)φn(x + iy) dx dy

y2

since sumW0(F) is a fundamental region for Γinfin But

1

int0φn(x + iy) dx = bny1ndashsn

where λn = sn(1 ndash sn) withreal sn ge 12 Thus

⟨θψ φn⟩ = bninfin

int0ψ(y)yndashsnndash1 dy

ergo θψ perp A0 Then by showing the dimension of δ(ξ)Coplus A1 is finite we see that the set

R =ψ φ isin Cinfin

00(R) supp ψ sub (0infin) θψ perp A

is quite large Furthermore it can be shown that the operator L sending f (y) 7rarr y2 f primeprime(y) hascontinuous spectrum over W = θψ ψ isin R and in fact E = W

Letting

Lψ(s) =infin

int0ψ(y)ysndash1 dy

an application of the Mellin transform to the function θψ yields

(37) θψ(z) =1

2πi int(σ)

Lψ(ndashs)E(z s χ) ds


where σ gt 1 and E(z s χ) is the Eisenstein series

(38) E(z s χ) = sumW0isinΓinfinΓ

χ(Wndash10 )(imageW0z

)s We postpone the discussion of the meromorphic continuation of Eisenstein series to sect41and take it for granted for the time being Pulling the line of integration to Δ = 1

2 we en-counter no poles and we obtain a transform of θψ(z) that involves E

(z 1

2 + it χ) These

Eisenstein series can be considered as lsquoeigenpacketsrsquo or lsquowave packetsrsquo making up the conti-nous spectrum They are themselves not in L2(ΓH χ) but after convolution with the enve-lope Lψ(s) they are Additionally the corresponding λ are with σ = 1

2 given by 14 + t2 and

hence belong to[14 infin

)

The incredible thing is that if one letss0 s1 sM

be the s-values (λ = s(1 ndash s) with

real sge 12) according to δ(χ)Coplus A1 (a set that is known to be finite) then

Proposition 314 ([23 Ch 6 Claim 95])(s0 s1 sM

)sub the poles of E(z s χ) where s0

exists iff δ(χ) = 1 In fact the poles of E(z s χ located inside real s gt 12 are simple and actually

coincide with(s0 s1 sM

)

With a little more work we arrive at the conclusion that the orthogonal complement of A1in A is exactly given by residues ress=sj E(z s χ) with (1ndashδ(χ)) le jleM These are residues ofthe Eisenstein series E(z s χ) We also arrive at the characterization of θψ(z) given in (37)This allows us to conclude that E is the closed subspace of L2(ΓH χ) generated by takingcontinuous superpositions of E

(z 1

2 + it χ)for 0 le t lt infin

This discussion has been non-rigorous but should serve as a conceptual guide to the tech-nical contortions required to put the decomposition (32) on solid ground Every claimabove can be justified but the justifications are far too long to be contained in this expo-sition

By way of a last remark in this subsection we note that in the case of Γ = SL(2Z) theset A1 is empty that is there is no contribution from residues of Eisenstein series The sen-sible reason for this is that in this case Es has one pole only in real sge 1

2 and it is simple withresidue a constant See [17 18] for a nice discussion of lsquoeigenpacketsrsquo arising from Eisen-stein series and a discussion of residues of Eisenstein series when Γ = SL(2Z) Also notethat when formulating the trace formula for SL(3Z) a case not treated here (among otherthings SL(3Z)SL(3R)SO(3R) is not a quotient of H = SL(2R)SO(2R)) residues ofEisenstein series very much do contribute cf [20 63ndash66 71]

33 The Selberg Trace Formula on ΓH In this subsection we develop the Selberg traceformula for the space L2(ΓH χ) where Γ is as in sect31 We are going to form an integraloperator of trace class and interpret it in two different ways to arrive at the trace formulaWe follow [23 Ch 6 sect10] We first consider the case χ(S) = 1

26 OWEN BARRETT

331 χ(S) = 1 In contrastwithbefore wewish to considerΦ isin C4(R+) such that∣∣∣φ(k)(t)

∣∣∣ leA(t+4)ndash1ndashkndashδ for 0 le kle4 and δ gt 0 Without loss of generality we may take Φ to be real Notethat Φ is not assumed to have compact support As before we introduce the point-pair in-variant

k(zw) = Φ(∣z ndash w∣2

image zimagew

)

and the kernel function


χ(σ)k(z σw)

It is not hard to show that k(zw) converges on H times H compacta and K(zw χ) rarr 0 asz rarr iinfin For a fixed w K(zw χ) expands as a Fourier series


Λ(λn)φn(z)φn(w)

which converges absolutely and uniformly on all of Htimes H Thanks to our choice of kernelwe have


infinsumn=1

Λ(λn)

It now comes down to attacking intFK(z z χ) dμ(z) After establishing the convergence of

intF

(sum∣∣tr(σ)∣∣=2

∣∣(k(z σz)∣∣ ) dμ(z)

the spectral decomposition of L2(ΓH χ) permits the assertion ([23 p102])

(39)

intF


∣∣(k(z σz)∣∣ ) dμ(z) = sum∣∣tr(σ)∣∣ =2χ(σ) int

Fk(z σz) dμ(z)

= sumσ

hyperbolic

χ(σ) intF[ZΓ(σ)]

k(z σz) dμ(z)

+ sumη

elliptic

χ(σ) intF[ZΓ(η)]

k(z ηz) dμ(z)

= sumσ

hyperbolic

χ(σ) logN(σ0)N(σ)12 ndash N(σ)ndash12

g(logN(σ))

+ sumη

elliptic

χ(η)2m(η) sin θ(η)

infin

intndashinfin

endash2θ(η)r

1 + endash2πrh(r) dr


Some words of explanation ZΓ(σ) is the centralizer of σ in Γ F[ZΓ(σ)] is again the funda-mental region of the subgroup ZΓ(σ) sub Γ N(σ) is the multiplier or characteristic constantof the hyperbolic element σ σ0 generates Z(σ) Last m(η) =

∣∣ZΓ(η)∣∣ and θ(σ) is defined

implicitly by the formula tr σ = 2 cos θ 0 lt θ lt π This work depends essentially on thesetting-up of the trace formula for vector-valued functions [22 p348ndash352] and for auto-morphic forms on a compact surface [22 p402ndash407]

In the case that ∣tr σ∣ = 2 we write wndash10 Skw0 where w0 isin ΓinfinΓ This coset decomposi-

tion is unique whenever σ = I It again follows from the convergence

intF

(sum

σ isin[S]∣tr σ∣=2

∣∣k(z σz)∣∣ ) dμ(z) lt infin

that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum

n=ndashinfinχ(Sn)k(z Snz)

)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum

n=ndashinfinχ(Sn)k(z Snz) dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(Sm)k(w0z Smw0z)

)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intsumw0(F)

(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT

The uniform convergence of sumσisinΓ∣∣k(z σw)

∣∣ on H times H compacta implies the continuity ofsumm =0 χ(Sm)k(z Smz) the series is also S-invariant We may conclude that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= Φ(0)μ(F) + intF([)Γinfin]

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0

χ(Sn)k(z Snz)) dx dy

y2

The second integral can be evaluated as follows ([23 p104ndash105])

infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1

Φ(n2t2 cos(2πnα))

dt

= 2 limεrarr0

infin

intε

( infinsumn=1

Φ(n2t2) cos(2πnα))

dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

It follows from summation by parts and trivial estimations that

2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du ≪ ε log

1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv

dv middot log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT

Combining the above we obtain


μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r


where all notation is as above We arrive at the final statement of the Selberg trace formulafor L2(ΓH χ) when χ(S) = 1

Theorem315 (The Selberg trace formula for L2(ΓH χ) when χ(S) = 1 [23 Ch 6Theorem105]) Suppose that h(r) satisfies the following hypotheses

(a) h(r) is analytic on∣∣image r∣∣ le 1

2 + δ fs δ gt 0(b) h(ndashr) = h(r)(c)∣∣h(r)∣∣≪ (

1 +∣∣real r∣∣)ndash2ndashδ

Suppose also that

g(u) =12π

infin

intndashinfin


and let χ be a 1 times 1 unitary representation of Γ satisfying χ(S) = 1 Introduce λn = 14 + r2n Then

infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr + g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



332 χ(S) = 1 In this section we follow [23 Ch 6 sect13] which in turn mostly fol-lows [56 pp 64ndash85] We would first like to unwind what we might hope for the devel-opment of Selbergrsquos trace formula for L2(ΓH χ) when χ(S) = 1 We recall the spectraldecomposition described in sect322 and stated in (32) namely

L2(ΓH χ) = δ(χ)Coplus L2cusp(ΓH) oplus L2


Imprecisely stated the naiumlve hope for obtaining a convergent useful trace formula wouldbe to try to lsquosubtractrsquo off the contribution of the trace of an integral operator arising fromthe continuous spectrum and thereby arrive at something well-behaved Since (cf [23(935) p91] for a nice enough f isin L2(ΓH χ) we should have following from the spectraldecomposition

(310) f (z) = sum cnφn(z) +infin

int0g(t)E

(z 1

2+ it χ

)dt

where cn = ⟨ f φn⟩ and g(t) =12π intF

f (z)E(

z 12

+ it χ)

dμ(z)

That is if k(zw) is a reasonable point-pair invariant and we form


χ(σ)k(z σw)

as usual then (310) implies that we should expect that

K(zw χ) = sum h(rn)φn(w)φn(z) +infin

int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)

It actually follows from Proposition 21 that

F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that

(311) K(zw χ) = sum h(rn)φn(w)φn(z) +12π

infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt

= sum h(rn)φn(w)φn(z) + H(zw χ) say

Our guess then is that the Selberg trace formula will result from an identity of the shape

intF

(K(z z χ) ndash H(z z χ)

)dμ(z) = sum h(rn)

32 OWEN BARRETT

It can be shown that if one writes down the decomposition (39) in the caseΦ(t) = (t+4)ndashγ

and χ equiv 1 one may show that the hyperbolic terms are well-behaved and

sum[σ]

hyperbolic

logN(σ0)N(σ)γ

lt infin forallγ gt 1

Anydifficulties in effecting the formula in this case then will arise from the parabolic termsUnfortunately handling these terms is quite thorny because of the growth properties ofK(zw χ) but everything works out in the end We summarize the results

Proposition 316 ([23 Ch 6 Prop 133]) For y v sufficiently largeK(zw χ) =

radicyvg(log sv

)+ O(1)

The implied constant depends only on Γ χ F and h

Proposition 317 ([23 Ch 6 Prop 134]) Assume h R rarr R is an even test function analyticand of sufficiently rapid decay (≪ endash7|r|) in the vertical strip

∣∣image r∣∣ le 1

2 + δ δ gt 0 LetK0(zw χ) = K(zw χ) ndash H(zw χ)

Let f isin Cinfin(H) cap L2(ΓH χ) satisfy Δ f +(14 + ξ2

)f = 0 overH Then

(i) K0(zw χ) isin C(Htimes H)(ii) K0(w z χ) = K0(zw χ)(iii) K0(σz ηw χ) = χ(σ)K0(zw χ)χ(η) for σ η isin Γ(iv) K0(zw χ) = O

((yv)1ndashβ

)for a small effective constant β for y v ≫ 1

(v) K0(zw χ) isin L2(F times F)(vi) intF K0(zw χ) f (w) dμ(w) = h(ξ) f (z) for z isin H and some ξ with

∣∣image ξ∣∣ le 1

2 + δ

Thefollowing is a crucial technical result that characterizes the L2 space and some importantintegral operators on it fairly completely

Proposition 318 ([23 Ch 6 Prop 136]) We suppose that(i) h(r) satisfies the assumptions of Proposition 317 and that moreover h(r) ≪ endash15|r|(ii) K0(zw χ) is defined as in Proposition 317(iii) L [ f ] = intFK0(zw χ)f(w) dμ(w) for f isin L2(ΓH χ)(iv) A is the closed subspace generated by the L2 eigenfunctions φn(v) E is the orthogonal complement of A and(vi) λn = sn(1 ndash sn) = 1

4 + r2nThen

(a) E is closed and L2(ΓH χ) = A oplus E (b) sum

(1 +∣∣rn))ndash4 lt infin


(c) sum endash14∣∣rn∣∣ ∣∣φn(z)φn(w)

∣∣≪ (yv)1ndashβ for y vge1 and β as in Proposition 317(d) sum h(rn)φn(z)φn(w) converges uniformly onHtimes H compacta(e) L [E ] = 0(f ) K0(zw χ) = sum h(rn)φn(z)φn(w) and(g) equation (311) is completely rigorous

With Proposition 318 in hand it is now more or less straightforward to obtain the Selbergtrace formula for L2(ΓH χ) We make a strong growth condition on the test function hnamely

h(r) ≪ endash15|r| for |image r| le 12

+ δ

We may can proceed roughly in analogy with sect331 see [23 p196]

sum h(rn) = intFK0(z z χ) dμ(z)

= intF


)dμ(z)

= intF

(sum

| tr σ|=2χ(σ)k(z σz) ndash H(z z χ)

)dμ(z)

+ intF

(sum

| tr σ| =2χ(σ)k(z σz)

)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r


= intFk(z z) dμ(z) + int

F

(sumn=0

k(z Snz) ndash H(z z χ))

dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1

0 Smw0)k(zwndash10 Smw0z)

)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(Sm)k(w0z Smw0z)

)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆

(noting that sumσisinΓ |k(z σw)| is uniformly convergent on H times H compacta summ =0 k(z Smz) is

continuous and S-invariant summ=0∣∣k (z Snz

)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆

where P is any region equivalent to sumw0 isin[S] w0(F) under Γinfin With some more work ([23p198ndash199]) to make everything explicit one obtains ([23 Eq 1320 p 207])

(312) sum h(rn) =μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr ndash12π intR

h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12

))ndash g(0) log 2 +

14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s

where w0 is understood asplusmn(

a bc d

) All the terms in Equation (312) are absolutely conver-

gent When Γ = PSL(2Z) and χ equiv 1

φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)

Γ(s)ζ(2s ndash 1)ζ(2s)

φ(s) is important because the it is closely related to the singularities of E(z s χ) It can beshown that

E(z s χ) = ys + φ(s)y1ndashs + O(radic

ω(t)e3|t|ndash2πy)

for y large

where ω is defined as follows See also Theorem 43 Lets0 s1 sM

be the eigenvalues

corresponding to δ(χ)CoplusA1 that is the poles of Eisenstein series located insidereal s gt 1

2

(The s0 entry is always omitted when δ(χ) = 0) Let

q = inf

x gt 0 sum|c|=x

χ(wndash1

00)= 0 w00 isin ΓinfinΓΓinfin

Form V(s) as

V(s) = q2sndash1φ(s)Mprodk=0

s ndash sk1 ndash s ndash sk

Then define

ω(r) = 1 ndashVprime (1

2 + ir)

V(12 + ir

) for r isin R

Evidently ω is a way of counting zeros of φ(s) and measuring the contribution to the spec-trum from the Eisenstein series

We emphasize

L2(ΓH χ) = A oplus E = sum[φn] oplus E

as in Proposition 318 We arrive at our final theorem

Theorem 319 (The Selberg trace formula for L2(ΓH χ) when χ(S) = 1 cf [23 Theorem138 p 209ndash210]) Suppose that h(r) satisfies the following hypotheses

(a) h(r) is analytic on |image r| le 12 + δ fs δ gt 0

(b) h(ndashr) = h(r) and(c) |h(r)| ≪

(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT

Suppose further that

g(u) =12π

infin

intndashinfin


Take χ(S) = 1 and introduce λn = sn(1 ndash sn) = 14 + r2n as in Proposition 318 Then with rn as in

Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin

h(r) tanh(πr) dr ndash12π intR

h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r


All sums and integrals converge absolutely Further we have the following version of Weylrsquos law

(i) 0le rn leR

+ 1

4π intRndashR

ω(r) dr sim μ(F)4π R2 and

(ii) ω(r) + φprimeφ(12 + ir

)= O(1)

Theorems 315 and 319 define lsquoversion Arsquo of the Selberg trace formula which includes thecase L2(PSL(2Z)H) = L2(PSL(2Z)H 1) (cf Table 1)

4 The Spectrum in Detail

In sect3 we considered a general discrete Fuchsian group Γ sub PSL(2R) whose fundamentalregion F had finite (non-Euclidean) area In this section and all that follows we will restrictto the primary case we care about the case of the arithmetic lattice Γ = PSL(2Z) Havingrecorded the genesis andmain rank-two development (for our purposes) of the Selberg traceformula in several compact and arithmetic cases we now proceed to enumerate some of theremarkable properties of the automorphic objects that constitute the spectrum L2(ΓH)

41 TheContinuous Spectrum Eisenstein series on ΓH We assume familiarity with thegeneralized upper half-plane and coset decomposition of H = SL(2R)SO(2R) See [20Ch 1] We remain in the rank 2 case We follow the exposition of [20 Ch 3 sect31] in thissection Define the function Is(z) = ys Is is an eigenfunction of the hyperbolic Laplacian


Δ = ndashy2(

partsup2partsup2x + partsup2

partsup2y

)with eigenvalue s(1ndashs) Ifreal sge 1

2 the function Is(z) is neither automor-

phic for SL(2Z) nor in L2(ΓH) (integration with respect to Haar measure) Perhaps themost naiumlve thing to do is to try to average Is(z) over the group to get an automorphic func-tion Since Is(z) is invariant under Γinfin we factor out by this subgroup Cosets ΓinfinSL(2Z)are determined like

Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1

Each relatively prime pair (c d) determines a coset This naiumlve idea produces an Eisensteinseries Compare to (38)

Definition 41 Let z isin Hreal s gt 1 Form the Eisenstein series

E(z s) = sumγisinΓinfinSL(2Z)

Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s

The followingproposition establishes that the serieswehave justwrittendown iswell-defined

Proposition 42 ([20 sect31 Prop 313]) The Eisenstein series E(z s) converges absolutely anduniformly on compact sets for z isin H andreal s gt 1 It is real-analytic in z and complex-analytic c Inaddition

(1) Fix ε gt 0 For σ = real sge1 + ε gt 1 there exists a constant c(ε) such that∣∣E(z s) ndash ys∣∣ le c(ε)yndashε for yge1

(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)

(3) ΔE(z s) = s(1 ndash s)E(z s)

Proof For yge1 we have∣∣E(z s) ndash ys∣∣le sum

(cd)=1cgt0

1c2σ

middot yσ∣∣∣z + dc

∣∣∣2σ= yσ sum

cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ

Since the set|z + (rc) + m| m isin Z 1 le rle c (r c) = 1

forms a set spaced by 1

c we maymajorize each term so that

ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT

fs ℓ isin Z There are at most φ(c) such terms for each ℓ It follows that

∣∣E(z s) ndash ys∣∣le yσ

infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ

le2yσ ζ(2σ ndash 1)ζ(2σ)

infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ

The second statement follows fromthe fact thatforallγ isin SL(2Z) γ(ΓinfinSL(2Z)) = (ΓinfinSL(2Z))The third follows from the eigenvalue of Is(z)

We now describe the Fourier expansion of E(z s)

Theorem43 ([20 p58 Theorem 318]) Letreal s gt 1 and z =( y x

0 1)isin H The Eisenstein series

E(z s) has the Fourier expansion

E(z s) = ys + φ(s)y1ndashs +2πsradicy

Γ(s)ζ(2s) sumn=0σ1ndash2s(n)|n|sndash

12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ

)is the Ramanujan sum

Proof First note that

ζ(2s)E(z s) = ζ(2s)ys + sumcgt0

sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0

and d = mc + r it follows that

ζ(2s)1

int0E(z s)e(ndashnx) dx

= ζ(2s)ysδn0 +infinsumc=1

cndash2scsumr=1

summisinZ

1

int0

yse(ndashnx)∣∣z + m + rc∣∣2s dx


cndash2scsumr=1

summisinZ

1+m+rc

intm+rc

yse(ndashn(x ndash rc))|z|2s

dx


c2scsumr=1

e(nr

c

) infin

intndashinfin

yse(ndashnx)(x2 + y2)s

dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1

int0E(z s)e(ndashnx) dx = ζ(2s)ysδn0 + σ1ndash2s(n)y1ndashs

infin

intndashinfin

e(ndashnxy)(x2 + 1)s

dx

where we understand σ1ndash2s(0) = ζ(1ndash2s) The theorem is now a consequence of the Fouriertransform identity

(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)

Γ(s) y = 02πs|y|sndash12

Γ(s) Ksndash 12(2π|y|) y = 0

This identity in turn is established by fact that endashπx2 is self-dual under Fourier transformapplied in the following way

Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin

endashundash2πixy( u1 + x2

)sdx du

u

=infin

int0endashuus

infin

intndashinfin

endashux2e(ndashxy) dx duu

=radic

πinfin

int0endashundashπ2y2uusndash12 du

u

Next we record the essential meromorphic continuation and functional equation of E(z s)this is used implicitly in many of the results of the previous section

40 OWEN BARRETT

Theorem 44 ([20 p59 Theorem 3110]) Let z isin H and s isin C with real s gt 1 The EisensteinseriesE(z s) and the functionφ(s) as in 41 can be continued tomeromorphic functions onC satisfyingthe functional equations

(1) φ(s)φ(1 ndash s) = 1(2) E(z s) = φ(s)E(z 1 ndash s)

The modified function Elowast(z s) = πndashsΓ(s)ζ(2s)E(z s) is regular except for simple poles at s = 0 1and satisfies the functional equation Elowast(z s) = Elowast(z 1 ndash s) The residue of the pole at s = 1 ( for allz isin H) is given by

ress=1 E(z s) =3π

It was Selberg [56 59] who first established the spectral theory and meromorphic continu-ation of Eisenstein series on GL(2) but Maaszlig who first made the Definition 41 Roelckestudied Eisenstein series attached to discrete groups other than SL(2Z) Selberg [58] estab-lished themeromorphic continuation of Eisenstein series for higher-rank groups and Lang-lands [36 37] brought this project to resolution in a pair of significant papers in which heestablished in themost general context themeromorphic continuation of Eisenstein seriesand complete spectral decomposition of arithmetic quotients ΓG where G is a reductivegroup and Γ is an arithmetic subgroup For more see Arthurrsquos exposition [2] of Eisensteinseries especially Langlandsrsquo work and the trace formula

42 The Discrete Cuspidal Spectrum Maaszlig forms on ΓH We turn now to the discretespectrum the Maaszlig forms on ΓH = SL(2Z)H In this section we follow [20 Ch 3] AMaaszlig form f ΓH rarr C is square-integrable that is

intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin

( dx dy)y2 is the invariantHaarmeasure on ΓH Maaszlig forms are automorphic objects thatis they are Γ-invariant hyperbolic eigenfunctions of the Laplacian More formally

Definition 45 ([20 sect33 Defn 331]) Let ν isin C A Maaszlig form of type ν for SL(2Z) is anon-zero function f isin L2(ΓH) that satisfies

(1) f (γz) = f (z)forallγ isin Γ z =( y x

0 1)isin H

(2) Δ f = ν(1 ndash ν) f and(3) int10 f (z) dx = 0 (cuspidality condition)

It is easy to show that if f is a Maaszlig form of type ν for SL(2Z) ν(1 ndash ν) is real and nonneg-ative and that if ν = 0 or 1 f is a constant function The first task is to develop a usefulFourier theory for Maaszlig forms for SL(2Z) The two key ingredients in this developmentare Whittaker functions onH and multiplicity one for such functions


421 Fourier-Whittaker expansion The fact that S =( 1 1

0 1)isin SL(2Z) means that f (z) is a

periodic function of x it must therefore admit a Fourier expansion of type

(43) f (z) = summisinZ

Am(y)e(mx)

Putting Wm(z) = Am(y)e(mx) by the absolute converges of the Fourier expansion Wm(z)must satisfy

ΔWm(z) = ν(1 ndash ν)Wm(z)Wm

(( 1 u0 1)middot z)= Wm(z)e(mu)

We say that Wm(z) is a Whittaker function of type ν associated to the additive charactere(mx) We state this implicit definition outright

Definition 46 A Whittaker function of type ν associated to an additive character ψ R rarr UwhereU = Ctimes(0infin) is the complex unit circle is a smooth nonzero functionW H rarr C satisfyingthe conditions

ΔW(z) = ν(1 ndash ν)W(z)W(( 1 u

0 1)middot z)= W(z)e(mu)

A Whittaker function W(z) of type ν and character ψ can always be written in the form

W(z) = Aψ(y) middot ψ(x)since the function W(z)φ(x) is invariant under translations x 7rarr x + uforallu isin R hence mustbe the constant function after fixing y

There are twomajor simplifications of crucial importance to theGL(2) theory multiplic-ity one and explicit realization of the one Whittaker function we can construct Namelysince we know that the function Iν(z) = yν (as in sect41) satisfies ΔIν(z) = ν(1 ndash ν)Iν(z) itis a good place to start if we want to actually write down a Whittaker function The othercondition is the basic observation that if h R rarr C is Cinfin and integrable and ψ R rarr Cis an additive character then the function H(x) = intinfinndashinfin h(u1 + x)ψ(ndashu1) du1 satisfies theequation H(u + x) + ψ(u)H(x) This follows from a simple change of variables

This statement plus the fact that since Δ is an invariant differential operator ΔIν(γz) =ν(1 ndash ν)Iν(γzforallγ isin GL(2R) means that the function

(44) W(z ν ψ) =infin

intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u

0 1)middot z)ψ(ndashu) du

=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT

is a Whittaker function of type ν and character ψOne fact that sets the case of GL(2) apart from the higher-rank cases is that W(z ν ψ)

can be expressed explicitly in terms of common special functions

Proposition 47 ([20 Prop 346 p 65]) Let ψm(u) = e(mu) and let W(z ν ψm) be theWhittaker function (44) Then

W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic

2πyKνndash 12(2π|m|y) middot e(mx)

where

Kν(y) =12

infin

int0endashy(u+1u)2uν du

u

is the classical K-Bessel function

Proof The naiumlve Fourier expansion (43) and the definition (44) imply that

W(z ν ψm) = W(y ν ψm) middot e(mx)

where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2

)νmiddot e(ndashum) du

= y1ndashνinfin

intndashinfin

e(ndashuym)(u2 + 1)ν

du

See (42) to complete the proof

The crucial ingredient in the Fourier theory of Maaszlig forms for SL(2Z) is the multiplicityone principle which does generalize to the higher-rank cases It is of such central impor-tance since it means that the one Whittaker function that we can construct is in fact theonly one we really need to construct

Theorem 48 (Multiplicity one [20 Theorem 348 p 66]) Let Ψ(z) be a Whittaker functionfor SL(2Z) of type ν = 0 1 associated to an additive character ψ which has rapid decay at the cuspThen

Ψ(z) = aW(z ν ψ)

for some a isin C with W(z ν ψ) given by (44) If ψ is trivial then a = 0

Proof sketch Using the classical theory of differential equations and the fact that Ψ(z) is aneigenfunction of the Laplacian it can be shown that Ψ(y) satisfies a differential equation


with exactly two linearly-independent solutions overC they areradic2π|m|y middot Kνndash 1

2(2π|m|y) andradic

2π|m|y middot Iνndash 12(2π|m|y)

where Iν and Kν are the classical I- and K-Bessel functions

Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and

Kν(y) =π2middot Indashν(y) ndash Iν(y)

sin πν=

12

infin

int0endashy(t+1t)2tν dt

t

The asymptotics

Iν(y) simey

radic2πy

Kν(y) simendashyradicπradic

2y

plus the growth condition on Ψ(z) obtain for us the theorem

We put together our explicit expression for W(z ν ψm) (Proposition 47) the naiumlve Fourierexpansion (43) and multiplicity one (Theorem 48) in the following proposition which isa completely explicit Fourier-Whittaker expansion of a Maaszlig form for SL(2Z)

Proposition 49 (Fourier-Whittaker expansions on GL(2R) [20 Prop 351 p 67]) Letf be a non-constant Maaszlig form of type ν for SL(2Z) Then for z isin H we have the Whittakerexpansion

f (z) = sumn=0

anradic

2π y middot Kνndash 12(2π|n|y) middot e(nx)

with complex coefficients an

Proof Beginning with the naiumlve Fourier expansion (43) ie

f (z) = sumnisinZ

An(y)e(nx)

and noting that it follows from Δ f = ν(1 ndash ν) f that

Δ(An(y)e(nx)) = ν(1 ndash ν)An(y)e(nx)we conclude that An(y)e(nx) must be aWhittaker function of type ν associated to e(nx) Theassumption that f is non-constant means that ν = 0 1 since f isin L2(ΓH) An(y)e(nx) musthave polynomial growth at the cusp The proposition now follows from themultiplicity oneTheorem 48

44 OWEN BARRETT

The mysteries of the discrete spectrum are myriad and devilishly hard For example ques-tions pertaining to

(a) the growth and distributional properties of Fourier coefficients of Maaszlig forms(b) the distribution of (L2 mass) of eigenfunctions of the Laplacian on themodular sur-

face(c) possible distributional properties admitted byMaaszlig form zeros on themodular sur-

face(d) automorphy of L-functions attached to symmetric powers symr f of a fixed Maaszlig

form

and many others are incredibly challenging In particular this is because the spectral andharmonic nature of these eigenfunctions resists interpretation in terms of algebra and coho-mology This is in blatant contrast to holomorphic cusp forms on GL(2) whose coefficientsadmit an algebraic interpretation that among other things allowed Deligne [11] to provebest-possible bounds on the growth of holomorphic cusp form Fourier coefficients Maaszligcusp form coefficients are conjectured to obey the same growth bound as holomorphic cuspform coefficients but entirely different tools will be necessary to prove the validity of thisconjecture which is known as the Ramanujan conjecture since it was Ramanujan whomadethe original conjecture for his tau function which appears in the Fourier expansion of thediscriminant cusp form the first holomorphic cusp form for the full modular group TheRamanujan conjecture is the topic of sect 43 where questions (a) and (d) are broached Ques-tion (b) is discussed in sect45 and question (c) is discussed in sect423

422 Hecke operators We would be remiss if we did not touch upon the Hecke theory asso-ciated to Maaszlig forms since without it we would be stuck with a far less coherent space ofmysterious automorphic forms with many bizarre properties In particular though we willnot delve into the L-theory ofHecke-Maaszlig forms the Euler product crucial to the formationof the L-function L(s f ) attached to a Hecke-Maaszlig cusp form would not exist if it were notfor the multiplicativity that the Hecke operators make transparent We follow [20 sect310ndash312]

We first formulateHecke operators in the the general setting of a groupG acting continu-ously on a topological space X Let Z sub Gbe a discrete subgroup Assume the quotient ZXhas a left Z-invariant measure dx and define the usual space L2(ZX) Let CG(Z) denote thecommensurator of Z defined as

CG(Z) =g isin G (gndash1Zg) cap Z has finite index in both Z and gndash1Zg


With d =[Z (gndash1Zg) cap Z

] we have the coset decompositions

Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi

which definesδidi=1 implicitly

The Hecke operators are then defined as follows

Definition 410 ([20 Defn 3105 p 75]) Let a group G act continuously on a topological spaceX and let Z be a discrete subgroup of G For each g isin CG(Z) we define a Hecke operator

Tg L2(ZX) rarr L2(ZX)

by the formula

Tg( f (x)) =dsumi=1

f (gδix)

forallf isin L2(ZX) x isin X

It is not hard to show that the operators Tg are well-defined ie theymap square-integrablefunctions to square-integrable functions It is easy to turn the set of Hecke operators intoa Z-module using the formula (mTg)( f ) = mTg( f ) for f isin L2(ZX) We can also define amultiplication operation on this module in the following way For g h isin CG(Z) considerthe coset decompositions

(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then

(ZgZ) middot (ZhZ) =cupj

ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ

The multiplication of Hecke operators for g h isin CG(Z) is then given by the formula

(46) TgTh = sumZwZsubZgZhZ

m(g hw)Tw

where

m(g hw) = i j Zαiβj = Zw

where αi βj are given by (45) This multiplication is associative and and so we may put aring structure on the set of Hecke operators per the following definition

46 OWEN BARRETT

Definition411 (TheHecke ring [20 Defn 3108 p 76]) LetG act continuously on a topolog-ical space X and let Z be a discrete subgroup of G Fix any semigroup Λ such that Z sub Λ sub CG(Z)The Hecke ring HZΛ is defined as the set of all formal sums

sumk

ckTgk

with ck isin Z gk isin Λ Multiplication in the ring is induced by (46)

Following [20 sect312] pass now to the specfic case of Z = Γ = SL(2Z) and X = H Forintegers n0 n1 ge1 the matrix

(n0n1 00 n0

)isin CG(Z) Let Λ denote the semigroup generated

by the matrices(

n0n1 00 n0

)for n0 n1 ge1 together with the modular group Γ

Theorem 412 The Hecke ring HΓΛ is commutative

For each nge1 we have a Hecke operator Tn It acts on the space of square-integrable auto-morphic forms f isin L2(ΓH) The action of Tn is given explicitly by the formula

Tn f (z) =1radicn sum

ad=n0le bltd

f(

az + bd

)

Theorem 413 Let ⟨middot middot⟩ denote the L2 (Petersson) inner product on L2(ΓH) The Hecke operatorsTn are self-adjoint with respect to ⟨middot middot⟩ ie

⟨Tn f g⟩ = ⟨ f Tng⟩ forallf g isin L2(ΓH)

Definition 414 Define the involution Tndash1 on L2(ΓH) byTndash1 f

(( y x0 1))

= f(( y ndashx

0 1))

Theorem 415 ([20 Theorem 3126 p 82]) The Hecke operatorsTnnisinN commute with each

other with the operator Tndash1 and with the Laplacian Δ = ndashy2(


partsup2y

)

It follows in a standard way that theHilbert space L2(ΓH) can be simultaneously diagonal-ized by the set of operators

T =Tn n = ndash1 n = 1 2

cupΔ

Therefore wemay considerMaaszlig forms that are simultaneous eigenfunctions of T We saysuch a form f is even if Tndash1 f = f odd if Tndash1 f = ndashf We let λn denote the eigenvalues of fwith respect to the Hecke operators ie

Tn f = λn f In some sense the lsquopointrsquo of the Hecke operators is that they provide additional structureon the space of Maaszlig forms and tell us something arithmetic about the Fourier coefficientsof Hecke-Maaszlig forms Namely we have the following essential theorem


Theorem 416 (Multiplicativity of Fourier coefficients [20 Theorem 3128 p83]) Consider

f(z) = sumn=0

a(n)radic

2πy middot Kνndash 12(2π|n|y) middot e(nx)

a Maaszlig form of type ν as in Proposition 49 which is also an eigenfunction of all the Hecke operatorsIf a(1) = 0 then f equiv 0 Assume f = 0 is normalized so that a(1) = 1 Then

Tn f = a(n) middot f foralln = 1 2

We have the following multiplicativity relations

a(m)a(n) = a(mn) if (m n) = 1

a(m)a(n) = sumd|(mn)

a(mn

d2

)

a(pr+1) = a(p)a(pr) ndash a(prndash1) forallp prime Z ni rge1

As already alluded to this multiplicative structure is necessary to develop the Euler productassociated to the L-function L(s f ) (This statement is a theorem ofHecke) The presence ofthe Hecke ring HΓΛ also has allowed for the resolution of the quantum unique ergodicityconjecture in the arithmetic case this is elaborated upon in sect45 Put neatly the Heckeoperators can be considered lsquoarithmetic symmetriesrsquo

423 Existence of Maaszlig forms amp nodal domains Before moving on to a discussion of theRamanujan conjecture in the next section we stop to make a short detour to illustrate howdifferent Maaszlig forms are from their holomorphic brethren Among other things while it iscompletely classical to write down cuspidal modular forms in terms of a q-expansion up tonow no one has found a single example of a Maaszlig form for PSL(2Z) though as discussedin sect44 Selberg used the trace formula that bears his name to prove that there are actuallyinfinitelymany Maaszlig [43] didwrite down some examples for discrete subgroups other thanPSL(2Z)

Another example of the difference in difficulty between the two kinds of cusp forms onGL(2) is found in the nature of their zeros on ΓH As a consequence of Dukersquos equidistribu-tion theorem [12] whichwe take up in detail in sect46 the zeros of holomorphic cusp forms onthe modular surface which are topologically discrete points become equidistributed in theappropriate limit SinceMaaszlig forms are actuallywaveforms eigenfunctions on themodularsurface which has dimension two as Riemannian manifold the nodes of a Maaszlig waveformf are actually a collection of nodal lines ndash classically lines on the modular surface that arelsquomotionlessrsquo As Ernsty Chladni developed a technique of picturing these lsquonodal linesrsquo on avibrating membrane we can inquire after what the nodal lines of a Maaszlig wave eigenformlook like on PSL(2Z)H ie we can try to make Chladni figures for Maaszlig forms In somesense this would be a lsquopicturersquo of a Maaszlig form

In some sense such a study would be a good study of the line between high-energy eigen-states in a classically chaotic system and classical chaos For one might hope to be able to

48 OWEN BARRETT

study how a chaotic system passes from lsquoclassical chaosrsquo to lsquoquantum chaosrsquo One way to dothis is to study high-energy eigenwaveforms These are theMaaszlig forms on ΓH The benefitof the arithmetic setting is that the additional number-theoretic symmetries afforded by theHecke operators make studying and computing with high-energy eigenwaveforms possible(or at least easier)

The existence of such waveforms in a more general setting than the modular surface saythe one considered previouslywherewe have a quotient of the upper half-plane by a possiblynonarithmetic discrete cofinite Fuchsian group with exactly one cusp is difficult to ascer-tain There is a general philosophy due to Phillips and Sarnak explicated in [48] whichinspired by work of Colin de Verdiegravere [9] on deformations or perturbations of a hyperbolicsurface relates the existence of cusp forms to the critical zeros of the Selberg zeta-functionZΓ(s) associated to a cofinite possibly non-arithmetic discete group Γ and in turn to thevanishing of an associated Rankin-Selberg L-function on the critical line The Phillips-Sarnak theory would suggest that in fact there are only finitely many waveforms on a gen-eral hyperbolic surface unless there is some additional structure and symmetry such as isavailable in the case Γ = PSL(2Z)

Putting aside this question which is in itself interesting we proceed to describe some re-sults ofHejhal andRackner [24] andGhosh Reznikov and Sarnak [19] onnodal domains ofMaaszligwaveforms in the arithmetic case of PSL(2Z)H We refer the reader to [24 Fig 6ndash8pp 285ndash287] for several remarkable lsquopicturesrsquo of nodal domains attached to various Maaszligwaveforms for PSL(2Z) The expectation is that the nodal lines will be come increasinglylsquochaoticrsquo or complex as the energy (eigenvalue) of the waveform gets large Indeed an un-derstanding of the interface between the classical and quantum chaos and the distributionof mass associated to waveforms on a general hyperbolic surface is the subject of so-calledquantum ergodicity statements taken up in sect45

We now summarize some quantitative results of Ghosh Reznikov and Sarnak [19] LetX denote the modular surface PSL(2Z)H We consider even Maaszlig forms onX and writethe eigenvalue associated to such a form f by λ f = 1

4 + t2f Let Z f sub X denote the nodal

line of f given byz isin X f (z) = 0

It consists of a finite union of real-analytic curves The

connected components ofXZ f are the nodal domains of f we denote their number byN f For any subsetY sub X letN Y

f denote the number of nodal domains associated to f that havenonempty intersectionwith Y Questions of interest toGhosh-Reznikov-Sarnak include (a)estimating N f and N Y

f and (b) determing whether Z f is non-singular ie whether or notthe nodal line intersects itself

Using a variety of heuristics arising fromphysicalmodels Bogolmony and Schmit conjec-ture [6] a precise asymptotic for N f in terms of the numbering of waveforms when orderedby eigenvalueenergy Let κ f denote the natural number associated to f by such an ordering


Conjecture 1 (Bogomolny-Schmit)

N f sim2π(3radic

3 ndash 5)κ f κ f rarr infin

Let σ denote the orientation-reversing isometry on X induced by the reflection of H givenby the involution σ(x+ iy) = ndashx+ iy The isometry σ maps the nodal domains of f bijectivelyto themselves a given nodal domain may either be unchanged after applying σ or it couldbe different Ghosh-Reznikov-Sarnak call a domain split or inert respectively We denotethe number of inert nodal domains by N f N f respectively By definition we have

N f = N f + N f

It is not known whether there are any split nodal domains Let d⋆z denote the arclength

measureradic

dx2+dy2y We now state some theorems of Ghosh-Reznikov-Sarnak

Theorem 417 ([19 Theorem 11]) Let C be a closed horocycle inX Then for ε gt 0

tndashεf ≪ε intC

f 2(z) d⋆z ≪ε tεf

This theorem clearly implies that C is not a part of Z f Let δ =z isin X σ(z) = z

Then

δ is an arc composed of three piecewise-analytic geodesics denoted by δ1 δ2 δ3 Let δ1 bethe sub-arc that is given by the projection toX of

z = x + iy isin H x = 0 yge1

and finally

let δ2 denote the sub-arc given by the projection of the geodesicz isin H x = 1 yge

radic3

2

Theorem418 ([19Theorem 13]) If β is a long enough but fixed compact subsegment in δ1 or δ2then for ε gt 0

1 ≪β intβ

f 2(z) d⋆zle intδ


Theorem 419 ([19 Theorem 14]) Let C be a fixed closed horocycle inX Then for ε gt 0

t112ndashεf ≪ε

∣∣∣Z f cap C∣∣∣≪ t f

Note that the random wave model of Bogolmony and Schmit predicts∣∣∣Z f cap C∣∣∣ sim length(C )

πt f

Theorem 420 ([19 Theorem 16]) Fix β sub δ a sufficiently long compact geodesic segment on δ1or δ2 and assume the Lindeloumlf Hypothesis for the L-functions L(s f ) Then∣∣∣Z f cap β

∣∣∣≫ε t112ndashεf

50 OWEN BARRETT

Theorem 421 ([19 Theorem 18]) With the same assumptions as in Theorem 420

N fβ ≫ε t

112ndashεf

and in particular that N βf goes to infinity as κ f does

Theproofs of these theorems use the arithmetic quantumunique ergodicity theoremof Lin-denstrauss [38] and Soundararajan [61]

43 TheRamanujanConjecture SrinivasaRamanujanwas interested in the arithmetic func-tion τ N rarr Z defined by the identity

sumnge1

τ(n)qn = q prodnge1

(1 ndash qn)24 = η(z)24 = Δ(z)

where q = e(z) with image z gt 0 η is the Dedekind eta function and Δ is the first cusp formie the unique weight 12 holomorphic cusp form for the full modular group It is known asthe discriminantmodular form From themodern viewpoint Ramanujan was investigatingthe Fourier coefficients of a holomorphic cusp form But by emphasizing τ as an arithmeticfunction Ramanujan observed a pair of properties of τ that prefigured the Hecke theoryand also conjectured that

∣∣τ(p)∣∣ le2p112 on primes p This growth estimate is now known

as the Ramanujan conjecture for holomorphic cusp forms In the modern setting we usu-ally normalize the coefficients by taking out a factor of n(kndash1)2 denoting a normalized τfunction with τ = τn112 Ramanujanrsquos original conjecture would be simply

∣∣τ(p)∣∣ le2Let g isin S⋆k(N) be a holomorphic cuspidal newform of weight k and level N g admits a

Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)

It is now classical that an = no(1) thanks to Delignersquos proof of the Weil conjectures includ-ing the Riemann hypothesis for function fields and his earlier work to connect the Weilconjectures to the growth of Fourier coefficients of holomorphic cusp forms Remarkablythe Satake parameters αp βp associated to the coefficient ap for p prime by αp +βp = ap (thecoefficients at primes determine the others completely via multiplicativity and the Heckerelations) have an interpretation as eigenvalues of Frobenius acting on ℓ-adic cohomologygroups The statement

∣∣αp∣∣ = 1 =

∣∣∣βp

∣∣∣ (for unramified primes p) then follows from thepurity theorem for eigenvalues of Frobenius In turn this gives the Deligne divisor bound

an le d(n)

where d the divisor function counts the number of divisors of n Note that d(n) = no(1)This achieves the achieves the Ramanujan conjecture for these holomorphic cusp forms


It would be reasonable to hope that the general growth bound an = no(1) holds more gen-erally for (appropriately-normalized) automorphic forms on GL(2) The appropriate gen-eralization for other Lie groups including groups of of higher rank was initially unclearSatake proposed [54] such a generalization but Kurokawa [34] and Howe and Piatetski-Shapiro [30] found that Satakersquos generalization did not hold A subsequent refinement ofthe conjecture by Piatetski-Shapiro [49] is the current formulation of the generalized Ra-manujan conjecture For an excellent discussion of the generalized Ramanujan conjecturesee Sarnak [53]

Aproblemalluded topreviously is establishing the automorphyof higher symmetric powerL-functions L(s symr f ) for Maaszlig forms on GL(2) Should the automorphy be obtainedfor all rge1 the Ramanujan conjecture would follow This is also known as establishingthe functoriality of symmetric powers of automorphic representations of GL(2) since inloose terms the principle of functoriality dictates that natural operations on automorphicL-functions carry over to operations on the side of automorphic representations Establish-ing that taking symmetric powers on the level of L-functions corresponds to an automorphicobject would be a functorial statement for symmetric powers of GL(2) It is then perhaps nosurprise that the current record for the best bound towards Ramanujan for GL(2) Maaszligforms were obtained by Kim and Sarnak as a consequence of Kimrsquos proof [33] of the au-tomorphy of the symmetric fourth see the second appendix They prove that the Satakeparameters αp βp associated to a Maaszlig form on GL(2) satisfy∣∣αp

∣∣ ∣∣∣βp

∣∣∣ le p764

Evidently for those familiar with GL(2) L-theory the bound on Fourier coefficients trans-lates into a statement about the abscissa of absolute convergence of the Dirichlet series as-sociated to L(s f ) for f a fixed Maaszlig cusp form

44 Weylrsquos Law As alluded to previously no one has written down an example of a Maaszligform for PSL(2Z) Any reasonable person might find this startling fact difficult to rec-oncile with how much has been proved about these forms Indeed one might expect thatautomorphic forms are too special to exist and worry that all these results are vacuous Tothe contrary it is a classical theorem of Selberg that in fact there are infinitely many Maaszligforms for SL(2Z) The gulf between this fact proven used the Selberg trace formula andthe fact that no one has been able to write down a concrete example indicates that it is likelythat Maaszlig forms are quite curious creatures and do not lend themselves to neat expressionin terms of say a compact q-expansion which is usually how modular forms are writtendown

We refer in part to the exposition ofMuumlller [47] Byway of introduction let us first returnto the case of a Riemann surface taken up in sect2 or more generally a smooth compactRiemannian manifold M of dimension n with smooth possibly empty boundary partM Thespectrum of the Laplacian on such an M is discrete as we have already discussed in the casen = 2 Theonly accumulation point isinfin and each eigenvalue occurswith finitemultiplicity

52 OWEN BARRETT

Ordering the eigenvalues of Δ on M like

0le λ0 le λ1 le middot middot middot rarr infin

we can count the number of eigenvalues less than some number R with multiplicity Let

N (R) = j radic

λj leX

denote such a count An obvious fundamental question is then how does N (R) grow withR The Weyl law in this setting states a precise asymptotic

N (R) sim vol(M)(4π)n2Γ

(n2 + 1

)Rn

It was proved by Weyl [68] in the case of a bounded domain in R3 Raleigh [50] for a cubeGarding [16] for a general elliptic operator on a domain in Rn and by Minakshisundaramand Pleijel [46] in the case of a closed Riemannian manifold A natural question then fol-lows what about the lower order terms in an asymptotic expansion of N (R) To addressthis form the remainder term

R(R) = N (R) ndashvol(M)

(4π)n2Γ(n2 + 1

)Rn

We then ask what is the optimal bound on the remainder For a closed Riemannian man-ifold Avakumović [4] proved the Weyl law with a power savings in the error term andshowed this bound of O(Rnndash1) for the error was optimal for the sphere This result wassubsequently extended by Houmlrmander [29] Weyl [69] then conjectured the exact form of anext-to-leading-order term in the case of a bounded domain Ω sub R3 precisely that

N (R) =vol(Ω)6π2 R3 ndash

vol(partΩ)16π

R2 + o(R2) R rarr infin

This statement was proved by Ivrii [31] and Melrose [45] for manifolds with boundary as-suming a certain condition on the period billiard trajectories

Wenowpass to considering the analogous question in the case ofX = PSL(2Z)H Usingthe trace formula he developed Selberg [55 (933) p668] proved the following version ofWeylrsquos law for an arbitrary lattice Γ in SL(2R)

NΓ(R) + MΓ(R) sim area(ΓH)4π

R2 R rarr infin

HereNΓ(R) counts the number of eigenvalues less thanRwithmultiplicity whileMΓ mea-sures the contribution from the continuous spectrum More precisely we have the statementalready stated inTheorem 319 (i) Combining it with item (ii) of the same theorem wemayconclude

(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin


Here φ is as given in (313) Note that this statement is valid in the case of a singular χ = 1This is precisely (up to normalization) Selbergrsquos statement [55 (933) p 668]

The basic issue with the Weyl law for ΓH as stated in (47) is that the contribution fromthe discrete spectrum is not isolated So we may have that NΓ(R) ≪ 1 and all the contri-bution comes from MΓ(R) Quoting Selberg [55 p 668] only in some special cases can thefunction φ(s) lsquobe expressed in terms of functions that are known from analytic number the-oryrsquo and in every one of these special cases MΓ(R) ≪ R log R Crucially for our intereststhese lsquospecial casesrsquo include the case of PSL(2Z) and congruence subgroups This showsthat there are infinitely many Maaszlig forms for PSL(2Z)H and the arithmetic congruencesubgroups we care about

Weylrsquos law for ΓH given in (47) constitutes an application that demonstrates the phe-nomenal power of the Selberg trace formula

Though we can separate out the contribution from the continuous spectrum and showthat it does not contribute to leading order in the arithmetic cases we care about a naturalquestion is could we somehow deform the arithmetic manifolds we care about and does theWeyl law still hold in such a lsquodeformedrsquo setting Any such investigation would be directlymotivated by the pioneering work of Phillips and Sarnak [48] as discussed in sect423 whostudy the real-analytic deformation of discrete groups in PSL(2R) The general Phillips-Sarnak philosophy is that the arithmetic structure present in for example theHecke struc-ture on ΓH that we discussed in sect422 is delicate in some rigorous sense That is to say itdoes not survive under deformation Following Phillips-Sarnak this structure is connectedto the zeros of the Selberg zeta-function along the critical line which is in turn related to thevanishing of an associated Rankin-Selberg L-function at these same points If these quan-tities varied continuously under deformation of the underlying manifold then the specialproperties we observe in the arithmetic case would be expected to break under deformationThis is essentially what Luo [40] shows Luo makes the notion of deformation rigorous inthe following sense Let Q isin S4(Γ0(p)) be a fixed newform where Γ0(p) is the usual congru-ence subgroup and p is prime And let T(Γ0(p)) be the associated Teichmuumlller space Thenhe considers the deformation Γτ generated by the quadratic differential Q(z) dz2 at Γ0(p)By showing that a positive proportion of Rankin-Selberg L-functions do not vanish at criti-cal zeros obtained from eigenvalues attached to an orthonormal basis of Hecke-Maaszlig cuspforms for the full modular group Γ0(1) Luo is able to prove in his Theorem 2 that the Weyllaw

NΓτ (R) sim area(ΓτH)4π

R2

fails for generic deformations Γτ under the assumption that the eigenvalue multiplicitiesof the Laplacian on Γ0(p)H are bounded (or a slightly weaker technical assumption) Thisremarkable result indicates that the arithmetic structure on Γ0(N)H is delicate in a rigoroussense

54 OWEN BARRETT

45 Quantum Ergodicity In sect423 we broached some of the remarkable properties ofMaaszlig forms on PSL(2Z)H In particular the nodal lines of Maaszlig forms suggested somekind of chaotic behavior as the energy of the waveform gets large We refer the readerto [41 52 62] for some critical developments in this area that inspire the following expo-sition

This section is devoted to the phenemenon of so-called quantum chaos or quantum er-godicity The study of quantum ergodicity is motivated by a desire to probe the boundarybetween classical and quantum models for a chaotic (ie regular or ergodic) system In par-ticular the goal is to understand how a quantum understanding of such a system is reflectedin how the system appears from a classical perspective

The above discussion seems like it might be of interest to physicists but what about itwould interest the number theorist The notion of arithmetic quantum chaos or arithmeticquantum ergodicity can be rephrased as a desire to understand the distribution of (L2)massof eigenfunctions of the Laplacian on arithmetic manifolds We pause to compare the twocases following [52]

Suppose Σg = π1(Σg)H is a compact hyperbolic surface where π1(Σg) sub PSL(2R) isdiscrete and cocompact The geodesic flow on the unit cotangent bundle is generated bya Hamiltonian and is known to be ergodic Anosov and appears chaotic The quantiza-tion of this Hamiltonian is the Laplace-Beltrami operator on Σg which we denote Δ Let φjdenote the eigenfunctions (Maaszlig forms) on Σg It would be natural to try to describe thedistributional properties of the φj Heller [25] found that in fact there was some structureassociated to these waveforms there was a concentration of L2 mass in certain states relatedto a finite union of periodic unstable orbits He called this phenomenon lsquoscarringrsquo

In an influential paper Rudnick and Sarnak [52] claim that this lsquoscarringrsquo is not genericMore precisely they define the probability measures μj on Σg by

dμj =∣∣∣φj(z)

∣∣∣2 dvol(z)

These probability measures have the quantum-mechanical interpretation the probabilitydensities of finding a particle in the state μj at the the point z Rudnick and Sarnak thenconjecture

Conjecture 2 (QuantumUnique Ergodicity) LetX be a compact manifold of negative curvatureThen the measures μj converge to dvol

The strength of this conjecture is that there is no need to pass to a subsequence of measuresμjk Rudnick-Sarnak conjecture that there is no exceptional subsequence of measures Asthey explain the physical interpretation is remarkable it is that lsquoat the quantum level and inthe semi-classical limit there is little manifestation of chaosrsquo ([52 p 196]) The uniquenessof the limit implies that there is only one quantum limit while classical unique ergodicitywhich is the uniqueness of the invariant measure for theHamiltonian flow is never satisfiedfor chaotic systems


At the time thatRudnick andSarnak conjecturedquantumunique ergodicity Schnirelman [60]Colin de Verdiegravere [10] and Zelditch [72] had proven that the quantum analogue of the ge-odesic flow on a compact Riemannian manifold Y is ergodic More specifically they lo-calize the measures μj to S⋆1(Y) the unit cotangent bundle and show that if the geodesicflow on S⋆1(Y) is ergodic there is a full-density subsequence λjk (where λj are the (ordered)eigenvalues of the spectrum of Δ on Y) for which μjk(A) rarr vol(A) vol(Y) for all suffi-ciently lsquonicersquo sets A such as geodesic balls (A full-density subsequence is one satisfyingsumλjk le λ 1 sim sumλj le λ 1) Zelditch [73] then extended this result to some noncompact sur-faces such as X = PSL(2Z)H As described by [41] he shows that if h isin Cinfin

00(Y) andintXh(x) dvol(x) = 0 then

sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ

Combining this with Selbergrsquos result [55] that sumλj le λ 1 sim λ12 establishes quantum ergodicity

in this setting The startling nature of the quantum unique ergodicity conjecture is that itmaintains there is no need to pass to a subsequence

In 1995 Luo and Sarnak [41] proved an analogue of quantumunique ergodicity for Eisen-stein series on X That is in analogy with the definition of the μj they formed the naturalmeasures2

μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)

and then showed that these lsquoEisensteinrsquo measures become individually distributed in thefull-sequence limit

Theorem 422 ([41 Theorem 11 p 208]) Let A B be compact Jordan-measurable subsets of XThen

limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)

In the general setting QUE remains open though Anantharaman [1] has showed that anylimitingmeasuremust have positive entropy In the case PSL(2Z)H however due to workof Lindenstrauss [38] and Soundararajan [61] the conjecture is proved We follow the ex-position of [62]

Lindenstraussrsquos approach to the problem is to consider microlocalizations of the Hecke-Maaszlig form measures μj to S⋆1(X) = SL(2Z)SL(2R) these lifts are approximately invari-ant under the geodesic flow from the results of Schnirelman [60] Colin de Verdiegravere [10]and Zelditch [72] as described to above Lindenstraussrsquos major contribution is to use re-sults from measure rigidity to show that the only possible limiting measures are of the form

2Note that μt(X) = infin so there is no natural normalization

56 OWEN BARRETT

3π c dx dy

y2 where 0 le cle1 This shows that the measures become equidistributed unless some

of the L2 mass lsquoescapesrsquo into the cusp Soundararajanrsquos major contribution was to show thatsuch lsquoescapersquo was impossible proving c = 1 in the statement above

Theorem 423 ([62 Theorem 1 p 359]) For any sequence of L2-normalized Hecke-Maaszlig eigen-forms φj the measures

∣∣∣φj

∣∣∣2 dx dyy2 tend weakly to the measure 3

πdx dyy2 as λj rarr infin

It is important to mention that the arithmetic structure afforded by the presence of theHecke ring on X was a crucial ingredient in the proof of Theorem 423 This is a sterlingexample of the value of the Hecke structure alluded to in previous sections SubsequentlyHolowinsky andSoundararajan [28] established anatural analogue ofQUE for holomorphicHecke eigencuspforms

A subconvexity bound for a particular degree-six L-function implies Theorem 423 viaWatsonrsquos explicit triple-product formula [67] This subconvexity bound follows in turn fromthe Lindeloumlf Hypothesis which is a consequence of the Generalized Riemann HypothesisSee Zhao [75] formore details In that paper Zhao studies the quantum variance as anotherangle on QUE More particularly he considers the sum

sumλj le λ

∣∣∣μj(ψ)∣∣∣2

where ψ isin Cinfin00(X) First introduced by Zelditch [74] the quantum variance describes the

variation in a quantum observable ⟨Op(ψ)μj μj⟩ = μj(ψ) where μj is the measure formedfrom a Hecke-Maaszlig form onX The quantum variance is related to the rate of convergenceto the limiting measure 3

πdx dyy2 in Theorem 423 Luo and Sarnak [42] consider the analo-

gous question for holomorphic cusp forms onX As a striking consequence it can be shownthat the zeros of holomorphic Hecke eigencuspforms of weight k become equidistributedon the modular surfaceX as k rarr infin see [51]

46 DukersquosTheorem In this final section the author indulges in discussing a favorite the-orem of his The setting is again the modular surfaceX = PSL(2Z)H We partially followthe exposition of [13] who give an alternate proof of Dukersquos theorem using ergodic theoryWe describe Dukersquos original proof which is closer in sprit to the focus of this expositionwhich is the spectral theory of automorphic forms and harmonic analysis

Dukersquos theorem [12 Theorem 1] is a statement about distribution of Heegner points andgeodesics on X We reproduce some of his background from [12 p 75] Denote by h(d)the number of proper (SL(2Z)-inequivalent) classes of primitive irreducible integral binaryquadratic forms Q = Q(x y) = ax2 + bxy + cy2 with discriminant d = b2 ndash 4ac h(d) is theclass number If d gt 0 Pellrsquos equation is x2 ndash dy2 = 4 Let (xd yd) with xd yd gt 0 be thefundamental solution and set εd = (xd +

radicd yd)2 Let Γ = PSL(2Z) and let F denote the


fundamental domain for ΓH If d lt 0 the h(d) Heegner points are given by

Λd =

zQ =

ndashb +radic

d2a

b2 ndash 4ac = d zQ isin F

If d lt 0 is fundamental then these points correspond to ideal classes inQ(radic

d) If d gt 0 thepoints

(ndashb plusmn

radicd)

2a specify the endpoints of a geodesic inHwith respect to the usual hy-

perbolicmetric ds2 = ( dx2+ dy2)y2 these endpoints induce a unique primitive positively-oriented closed geodesic in ΓH of length log εd or 2 log εd according as whether Q is or isnot equivalent to ndashQ If d gt 0 let Λd denote the set of all such distinct geodesics The totallength of geodesics in Λd is h(d) log εd and every primitive positive-oriented closed geo-desic in ΓH occurs in exactly one Λd We recall that d isin Z is a fundamental discriminant ifit isequiv 1 mod 4 and squarefree or equal to 4m where m equiv 2 or 3 mod 4 is squarefree Letdμ(z) = 3

πdx dyy2 be the familiar measure normalized so that μ(F) = 1 Duke proves

Theorem 424 ([12 Theorem 1 p 75]) Suppose d is a fundamental discriminant and Ω sub F isconvex (in the hyperbolic sense) and partΩ is piecewise-smooth Then for some δ gt 0 depending only onΩ

Λd cap ΩΛd

= μ(Ω) + O(|d|ndashδ

)as d rarr ndashinfin and(i)

sumCisinΛd

∣∣C cap Ω∣∣

sumCisinΛd

∣∣C ∣∣ = μ(Ω) + O(dndashδ)

as d rarr +infin(ii)

where∣∣C ∣∣ is the (non-Euclidean) length of C and the implied constants depend only on δ and Ω

though ineffectively

Part (i) with error term O(logndashA |d|

)for some A gt 0 with additional conditions on the

fundamental discriminant was proved by Linnik [39] using ergodic methods Log savingsare common in theorems proved using ergodic techniques though power savings like thoseDuke obtains are obviously preferable Chelluri [8] subsequently extended Theorem 424 tothe unit cotangent bundle S⋆1(X) he obtained the following There is a geodesic orbit asso-ciated to each geodesic class in Λd which can be lifted to the unit tangent bundle of H andthen projected to a geodesic orbit on S⋆1(X) Let Gd denote the image of Λd on S⋆1(X) Gd isa collection of compact orbits of the geodesic flow and it carries a natural probability mea-sure μd that is invariant under this geodesic flow Chelluri proves the following extension ofTheorem 424

Theorem 425 ([8]) As d rarr +infin amongst the positive fundamental discriminants the set Gd be-comes equidistributed with respect to the Liouville (Haar) probability measure μL on S⋆1(X) for any

58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd

ψ(t) dμd(t) rarr intS⋆1(X)

ψ(u) dμL(u)

Dukersquos theorem Theorem 424 is a beautiful statement about the equidistribution of closedgeodesics onX Note that if Gaussrsquo conjecture that h(d) = 1 for infinitelymany fundamentald gt 0 holds then by (ii) of Theorem 424 we have that in fact individual geodesics becomeequidistributed in the limit of large d

The proof of Theorem 424 is a beautiful application of Selbergrsquos spectral decompositionof L2(ΓH) Weylrsquos equidistribution criterion and a subconvexity estimate for Fourier coef-ficients of particular Maaszlig forms of half-integral weight We sketch it

Sketch of proof of Theorem 424 By Weylrsquos equidistribution criterion and the spectral decom-position of L2(ΓH) the theorem is proved if the following two lsquoWeyl sumsrsquo can be shownto decay in |d| The sums are

WEis(d t) =

1

Λd sumzisinΛdE(z 1

2 + it)

d lt 01

sum∣∣C ∣∣ sumCisinΛd intC E

(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1

Λd sumzisinΛdu(z) d lt 0

1sum∣∣C ∣∣ sumCisinΛd intC u(x) ds d gt 0

Here E(z s) is an Eisenstein series and u(z) is a certain automorphic eigenfunction of theLaplace-Beltrami operator on amatrix space There is a theta lift of u due toMaaszlig that yieldsaMaaszlig form of half-integral weight Let θ(z) denote Siegelrsquos theta function for an indefinitequadratic form as in [12 Theorem 3 p 81]

Theorem426 ([12Theorem4 p 84]) LetS[x]be an integral ternary quadratic formof signature(1 2) There is a subset of the integersDS sub Zdepending on S such that for f (z) = y34⟨u(middot) θ(z middot)⟩and d isin DS the dth Fourier coefficient of f (z) (at the cusp) is given by

ς(d)πndash sgn(d)4radic

2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0

sumCisinΛ+dintC u(z) ds d gt 0

Here Λ+d is a certain collection of points in ΓH The tick in the first sum indicates that u(z) is divided

by the order of the stabilizer of z in Γ


The point is that we can now use a subconvexity bound for Fourier coefficients of Maaszligforms that Duke proves using an estimate of Iwaniec for a certain sum of Kloosterman sumsover varying levels together with Proskurinrsquos generalization of the Kuznetsov sum formulaConcretely he proves

Theorem 427 ([12 Theorem 5 p 85]) Letς(n)

be the Fourier coefficients of a spectral Maaszligform f (z) of weight k = 12 + ℓ and (even) discriminant D for Γ0(N) where ℓ isin Z and N equiv 0mod D with eigenvalue λ = 14 + t2 We have the estimate

ς(n) ≪kDε |λ|A cosh(πt2)|n|ndash27+ε as |n| rarr infinprovide n is squarefree or a fundamental discriminant We may take A = 54 ndash k4 sgn(n)

Combining this estimate for Fourier coefficients of half-integral weight Maaszlig forms withMaaszligrsquos explicit theta correspondance we can estimate the Weyl sums Wcusp(d t) We alsoneed Siegelrsquos (ineffective) estimate in the forms

Λd ≫ε |d|12ndashε as d rarr ndashinfin and

sumCisinΛd

|C | ≫ε |d|12ndashε as d rarr +infin

Combining these estimates Duke shows for d a fundamental discriminant

WEis(d t) ≪ε |t|A|d|ndash14+εL(12 + it χd

)as |d| rarr infin

where A gt 0 is a constant that may change from line to line Theorem 427 then yields theestimate

WEis(d t) ≪ε |t|A|d|ndash128+ε

On the cuspidal side combining the above theorems and estimates wehave for fundamentald

Wcusp(d t) ≪ε |t|A|d|ndash128+ε

The theorem now follows in a standard way from the Weyl equidistribution criterion

References

[1] N Anantharaman Entropy and localization of eigenfunctions Ann of Math (2) 168 (2008) 435ndash475[2] J Arthur Eisenstein series and the trace formula (Oregon State University Corvallis OR 1977) Automor-

phic forms representations and L-functions 1 Proc Sympos Pure Math vol XXXIII Amer MathSoc Providence RI 1979 pp 253ndash274

[3] The work of Ngocirc Bao Chacircu Proc International Math Congr Hyderabad (2010) 1ndash14[4] V G Avakumović Uumlber die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten Math Z 65

(1956) 327ndash344[5] V Blomer Applications of the Kuznetsov formula on GL(3) Invent math 194 (2013) 673ndash729[6] E Bogomolny and C Schmit Percolation model for nodal domains of chaotic wave functions Physical Review

Letters 88 (2002)[7] D Bump Spectral theory of ΓSL(2R) (2001) lecture notes

60 OWEN BARRETT

[8] T Chelluri Equidistribution of roots of quadratic congruences 2004 Rutgers The State University of NewJersey New Brunswick PhD Thesis

[9] Y Colin de Verdiegravere Pseudo-Laplacians I 32 275ndash286 II 33 87ndash113 Ann Inst Fourier (1983)[10] Ergodiciteacute et functions propre du laplacian Commun Math Phys 102 497ndash502[11] P Deligne La conjecture de Weil II Inst Haute Eacutetudes Sci Publ Math 52 (1980) 137ndash252[12] W Duke Hyperbolic distribution problems and half-integral weightMaass forms Invent math 92 (1988) 73ndash

90[13] M Einsiedler E Lindenstrauss P Michel and A Venkatesh The distributino of closed geodesics on the mod-

ular surface and Dukersquos theorem LrsquoEnseignement Matheacutematique (2) 58 (2012) 249ndash313[14] N ElkiesTheRiemann zeta function and its functional equation (and a review of theGamma function andPoisson

summation) lecture notes available at wwwmathharvardedu~elkiesM25902zeta1pdf[15] P Garabedian Partial Differential Equations John Wiley 1964[16] L Garding Dirichletrsquos problem for linear elliptic partial differential equations Math Scand 1 (1953) 55ndash72[17] P Garrett Continuous spectrum for SL2(Z)H (2011) lecture notes available at http

wwwmathumnedu~garrettmmfmsnotes_ccont_afc_specpdf[18] Continuous spectrum for SL2(Z)H (2014) lecture notes available at http

wwwmathumnedu~garrettmmfmsnotes_2013-1413_1_cont_afc_specpdf[19] A Ghosh A Reznikov and P Sarnak Nodal Domains of Maass Forms I Geom Funct Anal 23 (2013)

1515ndash1568[20] D Goldfeld Automorphic Forms and L-Functions for the Group GL(nR) Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge UK 2006[21] G Halaacutesz Selbergrsquos trace-formula a generalization of the Poisson summation formula lecture notes available

at wwwmath-insthu~majorarticlesselberg1pdf[22] D Hejhal The Selberg Trace Formula for PSL(2R) Lecture Notes in Mathematics vol 1 Springer-Verlag

Berlin-Heidelberg-New York 1976[23] The Selberg Trace Formula for PSL(2R) Lecture Notes in Mathematics vol 2 Springer-Verlag

Berlin-Heidelberg-New York 1983[24] DHejhal andB RacknerOn theTopography ofMaassWaveforms forPSL(2Z) ExperMath 1 (1992) no 4[25] E J Heller Chaos and Quantum Physics (Les Houches 1989) (M J Giannoni A Voros and J Zinn-

Justin eds) Amsterdam North-Holland 1991 pp 549ndash661[26] E Hewitt and K Ross Abstract Harmonic Analysis Vol 1 Springer-Verlag Berlin 1963[27] D Hilbert Grundzuumlge einer allgemeinen Theorie der linearen Integralgleichungen BG Teubner Leipzig

1912[28] R Holowinsky and K Soundararajan Mass equidistribution for Hecke eigenforms Ann of Math (2) 172

(2010) 1517ndash1528[29] L Houmlrmander The spectral function of an elliptic operator Acta Math 121 (1968) 193ndash218[30] R Howe and I I Piatetski-Shapiro A counterexample to the ldquogeneralized Ramanujan conjecturerdquo for (quasi-

) split groups (Oregon State Univ Corvallis Ore 1977) Automorphic forms representations and L-functions 1 (A Borel and W Casselman eds) Proc Sympos Pure Math vol XXXIII Amer MathSoc Providence RI 1979 pp 315ndash322

[31] V Ivrii The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundaryFunktsional Anal i Prilozhen 14 (1980) no 2 24ndash34

[32] H Iwaniec Spectral Methods of Automorphic Forms 2nd ed Graduate Studies in Mathematics vol 53Amer Math Soc and Revista Matemaacutetica Iberoamericana Providence RI and Madrid 2002

[33] H Kim Functoriality for the exterior square of GL4 and symmetric fourth of GL2 with an appendix by DRamakrishnan and an appendix by H Kim and P Sarnak J Amer Math Soc 16 (2002) no 1 139ndash183

[34] N Kurokawa Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two Invent Math 49(1978) no 2 149ndash165


[35] E Lapid Introductory notes on the trace formula Automorphic Forms and the Langlands Program (L Ji KLiu S-T Yau and Z-J Zheng eds) Advanced Lectures in Mathematics vol IX 2010 pp 135ndash175

[36] R P Langlands Eisenstein series Algebraic Groups and Discontinuous Subgroups Proc Symp PureMath vol IX Amer Math Soc Providence RI 1966 pp 235-252

[37] On the functional equations satisfied by Eisenstein series Lecture Notes in Mathematics Springer-Verlag 1976

[38] E Lindenstrauss Invariantmeasures and arithmetic quantumunique ergodicity Ann ofMath (1)163 (2006)165ndash219

[39] Y V Linnik Ergodic properties of algebraic fields Springer Berlin-Heidelberg-New York 1968[40] W Luo Nonvanishing of L-values and the Weyl law Ann Math 154 (2001) 477ndash502[41] W Luo and P Sarnak Quantum ergodicity of eigenfunctions on PSL2(Z)H2 Inst Haute Eacutetudes Sci Publ

Math 81 (1995) 207ndash237[42] Quantum variance for Hecke eigenforms Ann Sci Eacutecole Norm Sup (4) 37 (2004) 769ndash799[43] HMaaszligUumlber eine neue Art von nichtanalytischen automorphen Funktionen und die BestimmungDirichletscher

Reihen durch Funktionalgleichungen Ann Math 122 (1949) 141ndash183[44] Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen Math Ann 125 (1953)

235ndash263[45] R BMelroseWeylrsquos conjecture formanifolds with concave boundary Geometry of the Laplace operator Proc

Sympos Pure Math vol XXXVI Amer Math Soc Providence RI 1980 pp 257ndash274[46] S Minakshisundaram and A Pleijel Some properties of the eigenfunctions of the Laplace-operator on Riemann-

ian manifolds Canadian J Math 1 (1949) 242ndash256[47] W Muumlller Weylrsquos law in the theory of automorphic forms Groups and Analysis The Legacy of Hermann

Weyl (K Tent ed) London Mathematical Society Lecture Note Series London Mathematical SocietyLondon 2008 2007 available at arxiv07102319[mathSP]

[48] R S Phillips and P Sarnak Invent Math 80 (1985) 339ndash364[49] I I Piatetski-ShapiroMultiplicity one theorems (Oregon State Univ Corvallis Ore 1977) Automorphic

forms representations and L-functions 1 (A Borel and W Casselman eds) Proc Sympos Pure Mathvol XXXIII Amer Math Soc Providence RI 1979 pp 209ndash212

[50] Lord Raleigh The dynamical theory of gases and of radiation Nature 72 (1905) 54ndash55 243ndash244[51] Z Rudnick On the asymptotic distribution of zeros of modular forms Int Math Res Not 34 (2005) 2059ndash

2074[52] Z Rudnick and P Sarnak The Behavior of Eigenstates of Arithmetic Hyperbolic Manifolds Commun Math

Phys 161 (1994) 195ndash213[53] P Sarnak Notes on the Generalized Ramanujan Conjectures Clay Math Proc 4 (2005)[54] I Satake Spherical functions and Ramanujan conjecture (Boulder Colo 1965) Algebraic Groups and Dis-

continuous Subgroups (A Borel and G Mostow eds) Proc Sympos Pure Math vol IX Amer MathSoc Providence RI 1966 pp 258ndash264

[55] A Selberg Harmonic analysis (1954) preprint Universitaumlt Goumlttingen reprinted in A Selberg CollectedPapers Vol 1 Springer-Verlag Berlin-Heidelberg-New York 1989

[56] Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applicationsto Dirichlet series J Ind Math Soc 20 (1956) 47ndash87

[57] Automorphic functions and integral operators Seminars on Analytic Functions Vol 2 Institute forAdv Study 1957 pp 152ndash161

[58] On discontinuous groups in higher dimensional symmetric spaces International Colloquium on Func-tion Theory Tata Institute Bombay 1960 pp 147ndash164

[59] Discontinuous groups and harmonic analysis Proc International Math Congr Stockholm (1962)177ndash189

[60] A Schnirelman Usp Mat Nauk (6) 29 (1974) 181ndash182

62 OWEN BARRETT

[61] K Soundararajan Quantum unique ergodicity for SL2(Z)H Annals of Math (2) 172 (2010) 1529ndash1538[62] Quantum Unique Ergodicity and Number Theory Proc International Math Congr Hyderabad

(2010) 357ndash382[63] D IWallaceMaximal parabolic terms in the Selberg trace formula for SL(3Z)SL(3R)SO(3R) J Number

Theor (2) 29 (1988) 101ndash117[64] Terms in the Selberg trace formula for SL(3Z)SL(3R)SO(3R) associated to Eisenstein series coming

from a maximum parabolic subgroup Proc Amer Math Soc (4) 106 (1989) 875ndash883[65] Terms in the Selberg trace formula for SL(3Z)SL(3R)SO(3R) associated to Eisenstein series coming

from a minimal parabolic subgroup Trans Amer Math Soc (2) 327 (1991) 781ndash793[66] The Selberg trace formula for SL(3Z)SL(3R)SO(3R) Trans Amer Math Soc (1) 345 (1994)

1ndash36[67] TWatsonCentralValue ofRankinTripleL-function forUnramifiedMaassCuspForms PrincetonPhD thesis

2004[68] HWeylDas asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer

Anwendung auf die Theorie der Hohlraumstrahlung) Math Ann 71 (1912) 441ndash479[69] Uumlber die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze Math Ann 71

(1913) 441ndash479[70] K Yosida Functional Analysis 2nd ed Springer-Verlag Berlin 1968[71] D Zagier Eisenstein series and the Selberg trace formula I Automorphic Forms Representation Theory and

Arithmetic Springer-Verlag Berlin-Heidelberg-New York 1981 pp 275ndash301[72] S Zelditch Uniform distribution of eigenfunctions on compact hyperbolic surfaces Duke Math J 55 (1987)

919ndash941[73] Selberg trace formulaelig and equidistribution theorems Memoirs of the AMS 96 (1992) no 465[74] On the rate of quantum ergodicity Commun Math Phys 160 (1994) 81ndash92[75] P Zhao Quantum Variance of Maass-Hecke Cusp Forms Commun Math Phys 297 (2010) 475ndash514

1 Introduction



31 Introducing the Selberg Trace Formula on Γ0C5C30F H

32 The Spectral Decomposition of Γ0C5C30F H

33 The Selberg Trace Formula on Γ0C5C30F H


41 The Continuous Spectrum Eisenstein series on Γ0C5C30F H

42 The Discrete Cuspidal Spectrum Maaszlig forms on Γ0C5C30F H

43 The Ramanujan Conjecture

44 Weyls Law

45 Quantum Ergodicity

46 Dukes Theorem

References

2 OWEN BARRETT

1 Introduction

The goal of this exposition is to discuss the spectrum of the Laplace-Beltrami operator Δon (compact and arithmetic) manifolds given by quotients of the upper half-planeH = z isinC imagez gt 0 and to introduce themain tool available to study the structure of the spectrumthe Selberg trace formula

On a compact hyperbolic surface S Δ can be interpreted as the quantum Hamiltonian ofa particle whose classical dynamics are described by the (chaotic) geodesic flow on the unittangent bundle of S The spectrum of Δ on S resists straightforward characterization andindeed it is only by considering the spectrum all at once that Selberg was able to find theidentity that bears his name He called this identity a lsquotrace formularsquo since it was obtainedby computing the trace of an integral operator on S of trace class in two different ways Thismethod generalizes the Poisson summation formula which is obtained in precisely the sameway In every case in which a trace formula exists the formula exploits the duality affordedby Fourier theory to relate on average a collection of spectral data to a collection of geo-metric data

An understanding of the spectrum and eigenfunctions of the Laplacian on a compactRiemannianmanifold S is of interest to physicists For example a basic problem that probesthe interaction between classical and quantum mechanics is the question of whether or notthe L2 mass of high-energy eigenfunctions becomes equidistributed over S in the limit Itis known that when the geodesic flow on S (corresponding to the classical dynamics) is er-godic such equidistribution is achieved generically we call such equidistribution quantumergodicity We would also like to know whether the genericity hypothesis may be lifted ifnot there would be a subset of lsquobadrsquo eigenfunctions that resist equidistribution If we needno assumption on genericity and all eigenfunctions equidistribute in the limit of high en-ergy we say the system exhibits quantum unique ergodicity This phenomenon and relatedconjectures are discussed in sect45

Automorphic forms and their associated L-functions are principal objects of study bymany contemporary number theorists Perhaps the most classical motivation may be foundin the connection between modular forms and elliptic integrals or modular functionsrsquo re-lationship to generating functions for combinatorial and number-theoretic quantities ofwhich the connection between the partition function and the Dedekind eta function is butone example More recently connections have been made between the coefficients of cer-tain modular forms and the representation theory of some sporadic groups In contempo-rary number theory modularity of elliptic curves has connected holomorphic cusp formsto elliptic curves and to the theory of Galois representations The automorphy of Galoisrepresentations or how to obtain a Galois representation from an automorphic form arequestions that are of central interest to number theorists today Statements relating to spe-cial values ofmotivic L-functions some of which are theorems likeDirichletrsquos class numberformula most of which are conjectures among them the Birch and Swinnerton-Dyer con-jecture and more generally the equivariant Tamagawa conjecture have provided clear evi-dence that global information of the utmost importance is encoded in L-functions attached






(s symr f



4 OWEN BARRETT





f(ξ)= int



summisinZ

f (m) = sumnisinZ

f (n)


g(y) = summisinZ

f (y + m)




R( f )φ(x) = intR



= intTsum


TK f (x y)φ(y) dy



nisinZf (n)



f (n)



θ(z) = sumnisinZ



sumnisinZ


endashπn2y


θ(1z) =radic




u real s gt 0


ξ(s) +1s+

11 ndash s

=12

infin


duu

6 OWEN BARRETT







Δ = y2(

partsup2partsup2x

+partsup2partsup2y

)


(21) D =1

ac ndash b2

[partpartξ


ac ndash b2

)+

partpartη


ac ndash b2

)]



μn = infin



infinsumk=1

1μ2k

lt infin







n=0

[φn]


(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)







(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)


intH



8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References






(s symr f



4 OWEN BARRETT





f(ξ)= int



summisinZ

f (m) = sumnisinZ

f (n)


g(y) = summisinZ

f (y + m)




R( f )φ(x) = intR



= intTsum


TK f (x y)φ(y) dy



nisinZf (n)



f (n)



θ(z) = sumnisinZ



sumnisinZ


endashπn2y


θ(1z) =radic




u real s gt 0


ξ(s) +1s+

11 ndash s

=12

infin


duu

6 OWEN BARRETT







Δ = y2(

partsup2partsup2x

+partsup2partsup2y

)


(21) D =1

ac ndash b2

[partpartξ


ac ndash b2

)+

partpartη


ac ndash b2

)]



μn = infin



infinsumk=1

1μ2k

lt infin







n=0

[φn]


(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)







(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)


intH



8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

4 OWEN BARRETT





f(ξ)= int



summisinZ

f (m) = sumnisinZ

f (n)


g(y) = summisinZ

f (y + m)




R( f )φ(x) = intR



= intTsum


TK f (x y)φ(y) dy



nisinZf (n)



f (n)



θ(z) = sumnisinZ



sumnisinZ


endashπn2y


θ(1z) =radic




u real s gt 0


ξ(s) +1s+

11 ndash s

=12

infin


duu

6 OWEN BARRETT







Δ = y2(

partsup2partsup2x

+partsup2partsup2y

)


(21) D =1

ac ndash b2

[partpartξ


ac ndash b2

)+

partpartη


ac ndash b2

)]



μn = infin



infinsumk=1

1μ2k

lt infin







n=0

[φn]


(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)







(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)


intH



8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References



R( f )φ(x) = intR



= intTsum


TK f (x y)φ(y) dy



nisinZf (n)



f (n)



θ(z) = sumnisinZ



sumnisinZ


endashπn2y


θ(1z) =radic




u real s gt 0


ξ(s) +1s+

11 ndash s

=12

infin


duu

6 OWEN BARRETT







Δ = y2(

partsup2partsup2x

+partsup2partsup2y

)


(21) D =1

ac ndash b2

[partpartξ


ac ndash b2

)+

partpartη


ac ndash b2

)]



μn = infin



infinsumk=1

1μ2k

lt infin







n=0

[φn]


(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)







(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)


intH



8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

6 OWEN BARRETT







Δ = y2(

partsup2partsup2x

+partsup2partsup2y

)


(21) D =1

ac ndash b2

[partpartξ


ac ndash b2

)+

partpartη


ac ndash b2

)]



μn = infin



infinsumk=1

1μ2k

lt infin







n=0

[φn]


(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)







(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)


intH



8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References







n=0

[φn]


(23) R( f )(z) = intΣg

k(zw) f (w) dμ(w)







(24) k(zw) = Φ(

|z ndash w|2

imagezimagew

)


intH



8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

8 OWEN BARRETT



k(σzw) zw isin H










n=0 Λ(λn)



Σg



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)





tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References



tr R =infinsumn=0

Λ(λn)= int

FK(z z) dμ(z)

= intF

sumσisinπ1(Σg)

k(σz z)

dμ(z)

= sumσisinπ1(Σg)

intFk(σz z) dμ(z)

= sum[σ]

distinct

sumRisin[σ]

intFk(Rz z) dμ(z)

= sum[σ]

distinct



= sum[σ]



= sum[σ]


intρ(F)

k(σξ ξ) dμ(ξ)


ρ(F)


tr R = sum[σ]

distinct

intF[ZΓ(σ)]

k(σz z) dμ(z)



10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

10 OWEN BARRETT



E = intF[ZΓ(σ)]


k(σz z) dμ(z)


k(σηw ηw) dμ(w)





(σ0)lt infin

Then






intF[ZΓ(σ)]



where

g(u) =infin

intx

Φ(t)radict ndash x







Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References



Φ(0) =14π

infin

intndashinfin

r( infin

intndashinfin

g(u)eiru du)

tanh(πr) dr


inf

ν ge 1

infinsumn=1


= 1






λn where


λn = ndashinfin



2 + irn we have

infinsumn=0

h(rn)

=μ(F)4π

infin

intndashinfin

r h(r) tanh(πr) dr

+ sum[σ]


g(logN(σ))

where

g(u) =12π

infin

intndashinfin



12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

12 OWEN BARRETT



sumγ


12π

infin

intndashinfin

h(r)Γprime

Γ(14 + 1

2 ir)

dr

ndash 2infinsumn=1

Λ(n)radicn

g log n


g(u) =infin

intndashinfin

g0(u + t)g0(t) dt




Γ(N) =(

a bc d



Γ0(N) =(

a bc d

) c equiv 0 mod N



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References



Γ1(N) =(

a bc d



∣∣ = sumd|N

φ(gcd(d Nd))









L2(GH χ) =






14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

14 OWEN BARRETT





Δk = y2(

partsup2partsup2x

+partsup2partsup2y

)ndash iky part

partx







( 1 10 1)












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References












χ(σ)k(z σw)


16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

16 OWEN BARRETT



n=0

[φn]







infinsumn=0

h(rn)







K(z z χ) =infinsum


infinsum

n=ndashinfinΦ(

n2

y2

)ge lfloory

radicαrfloor

In particularinfin

intY

1

int0K(z z χ) dx dy



(31) δ(χ) =












cont(SL(2Z)H)











where w = reiθ =



18 OWEN BARRETT

Define






4








ndashn2

r2





4πg0(r)





4πΓ(2s)


∣∣∣2)sF(


∣∣∣2)







sumσisinΓ









Gs(z z0 χ) =12π

log∣∣z ndash z0

∣∣ + O(1) near z = z0


Gs(z z0 χ) =




20 OWEN BARRETT


sumσisinΓ





1 ndash 2sndash sum

n+α =0Fn(w s χ)y

12Ksndash 1

2

(2π ∣n + α∣ y) e((n + α)z)


1 ndash 2s+ O



χ(Wndash1

0)image(W0w)

12 Isndash 1

2








y12 Isndash 1

2(2π|n + α|y)e((n + α)z)

ys




(34) intF














22 OWEN BARRETT




K(s t) =κ(s t)rℓ

0le ℓ lt dimD






f (s) =infinsumn=1

(φn f

)φn(s) =

infinsumn=1

(φn g

)λn

φn(s)





n=1

[φn]













amy12Ksndash 1

2(2π|m|y)e(mx)



A0 =oplus

0ltλltinfinA0(λ)

A1 =oplus

0ltλlt 14

A1(λ)





(δ(χ)Coplus A1



24 OWEN BARRETT



χ(Wndash1

0)ψ(imageW0z

)



W0

χ(Wndash10 ) int



= sumW0

intW0(F)


=infin

int0

1


y2


1









Letting

Lψ(s) =infin



(37) θψ(z) =1

2πi int(σ)








(z 1

2 + it χ) These




)








)


(z 1






26 OWEN BARRETT




image zimagew

)



χ(σ)k(z σw)



Λ(λn)φn(z)φn(w)



infinsumn=1

Λ(λn)


intF


∣∣(k(z σz)∣∣ ) dμ(z)


(39)

intF



Fk(z σz) dμ(z)

= sumσ

hyperbolic


k(z σz) dμ(z)

+ sumη

elliptic


k(z ηz) dμ(z)

= sumσ

hyperbolic


g(logN(σ))

+ sumη

elliptic


infin

intndashinfin

endash2θ(η)r








intF

(sum



that

intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intF

(sum

m =0χ(wndash1

0 Smw0)k(zwndash1

0 Smw0z))

dμ(z)

= intF

( infinsum


+ sumw0 isin[S]

intF

(sum


)dμ(z)

= intF

( infinsum


)dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)

= Φ(0)μ(F) + intF

(sumn=0

χ(Sn)k(z Snz))

dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0χ(Sm)k(w Smw)

)dμ(w)


(sum

m=0χ(Sm)k(z Smz)

)dμ(z)

28 OWEN BARRETT



intF

(sum∣tr σ∣=2

χ(σ)k(z σz))

dμ(z)


(sumn=0

χ(Sn)k(z Snz))

dμ(z)

= Φ(0)μ(F) +infin

int0

1

int0

(sumn =0


y2


infin

int0

1

int0

(sumn =0


y2

=infin

int0

1

int0

(sumn=0

Φ(

n2

y2

)e(nα)

) dx dyy2

= 2infin

int0

( infinsumn=1

Φ(

n2

y2

)cos(2πnα)

) dyy2

= 2infin

int0

( infinsumn=1


dt

= 2 limεrarr0

infin

intε

( infinsumn=1


dt

Now

2infin

intε

( infinsumn=1


dt

= 2infinsumn=1

cos(2πnα)infin

intεΦ(n2t2) dt

= 2infinsumn=1

cos(2πnα)n

infin

intnε

Φ(u2) du

= 2infinsumn=1

cos(2πnα)n

( infin

int0Φ(u2) du ndash

nε

int0Φ(u2) du

)


= 2infin

int0Φ(u2) du

infinsumn=1

cos(2πnα)n

ndash 2infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du

= 2infin

int0Φ(u2) du log

∣∣∣∣ 11 ndash e(α)

∣∣∣∣ndash 2

infinsumn=1

cos(2πnα)n

nε

int0Φ(u2) du


2infinsumn=1

cos(2πnα)n

nε


1ε

+ ε

and hence that

2infin

intε

( infinsumn=1


dt

=infin

int0

Φ(v)radicv


∣∣∣∣ + O(ε log

1ε

)= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣ + O(ε log

1ε

)

and

infin

int0

1

int0

(sumn=0


y2= g(0) log

∣∣∣∣ 11 ndash χ(S)

∣∣∣∣

30 OWEN BARRETT



μ(F)4π

infin

intndashinfin

rh(r) tanh(πr) dr

+ g(0) log∣∣∣∣ 11 ndash χ(S)

∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r







Suppose also that

g(u) =12π

infin

intndashinfin



infinsumn=1

h(rn) =μ(F)4π

infin

intndashinfin


∣∣∣∣+ sum

[σ]hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r








int0g(t)E

(z 1

2+ it χ

)dt


f (z)E(

z 12

+ it χ)

dμ(z)



χ(σ)k(z σw)



int0F(t)E

(z 1

2+ it χ

)dt

where

F(t) =12π intF

K(zw χ)E(

z 12

+ it χ)

dμ(z)


F(t) =12π int

H

k(ξw)E(ξ

12

+ it χ)

dμ(ξ) =12π

h(t)E(

w12

+ it χ)

and therefore that


infin

int0h(t)E

(w

12

+ it χ)E(

z 12

+ it χ)

dt



intF


)dμ(z) = sum h(rn)

32 OWEN BARRETT



sum[σ]

hyperbolic

logN(σ0)N(σ)γ




radicyvg(log sv

)+ O(1)






)f = 0 overH Then


((yv)1ndashβ




2 + δ



4 + r2nThen








+ δ



= intF


)dμ(z)

= intF

(sum


)dμ(z)

+ intF

(sum


)dμ(z)

= intF

(sum


)dμ(z)

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



F

(sumn=0


dμ(z)

+ intF

(sum

σ isin[S]|trσ|=2

χ(σ)k(z σz))

dμ(z) + ⋆

34 OWEN BARRETT

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum

m=0χ(wndash1


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intF

(sum


)dμ(z) + ⋆

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ sumw0 isin[S]

intw0(F)

(sum

m=0k(w Smw)

)dμ(z) + ⋆



)∣∣ = O(y2+2δ

)for 0 lt y lt 1)

= Φ(0)μ(F) + intF

(sumn=0


dμ(z)

+ intP

(sum

m=0k(w Smw)

)dμ(w) + ⋆


(312) sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π intR

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



where

(313) φ(s) =radic

πΓ(s ndash 1

2)

Γ(s) sumw0 isin[S]

χ(wndash1

0) 1|c|2s


a bc d



φ(s) =radic

πΓ(s ndash 1

2)

Γ(s)

infinsumk=1

φ(k)k2s

=radic

πΓ(s ndash 1

2)





for y large


be the eigenvalues


2


q = inf

x gt 0 sum|c|=x

χ(wndash1


Form V(s) as



Then define


2 + ir)

V(12 + ir

) for r isin R


We emphasize






(1 + |real r|

)ndash2ndashδ

36 OWEN BARRETT


g(u) =12π

infin

intndashinfin



Proposition 318

sum h(rn) =μ(F)4π

infin

intndashinfin


h(r)Γprime(1 + ir)

Γ(1 + ir)dr

+14h(0)

(1 ndash φ

(12


14π

h(t)φprime(12 + it

)φ(12 + it

) dt

+ sum[σ]

hyperbolic


g(logN(σ))

+ sum[η]

elliptic


infin

intndashinfin

endash2θ(η)r



(i) 0le rn leR

+ 1

4π intRndashR



)= O(1)






Δ = ndashy2(


partsup2y




Γinfin

(a bc d

)=(

lowast lowastc d

)=(

u ndashvc d

) du + cv = 1




Is(γz)2

=12 sum

cdisinZ(cd)=1

ys

|cz + d|2s




(2) E(

az+bcz+d s

)= E(z s) for all

(a bc d

)isin SL(2Z)



(cd)=1cgt0

1c2σ



cge1

csumr=1

(rc)=1

1c2σ sum

misinZ

1∣∣z + rc + m

∣∣2σ




ℓ le∣∣∣x +

rc+ m

∣∣∣ lt ℓ + 1

38 OWEN BARRETT



infinsumc=1

φ(c)c2σ sum

ℓisinZ

1(ℓ2 + y2)σ


infinsumℓ=0

1(ℓ2 + y2)σ


(yndash2σ +

infin

int0

du(u2 + y2)σ

)≪ y1ndashσ








12Ksndash 1

2(2π|n|y)e(nx)

where

(41) φ(s) =radic

πΓ(s ndash 1

2)


σs(n) = sumd|ndgt0

ds

Ks(y) =12

infin

int0endash

12 y(u+1u)us du

u and

S(n c) =csumr=1

(rc)=1

e(nr

c

)= sum

ℓ|nℓ|c

ℓμ( cℓ




sumdisinZ

ys

|cz + d|2s


Letting δn0 =

1 n = 00 n = 0


ζ(2s)1



cndash2scsumr=1

summisinZ

1

int0



cndash2scsumr=1

summisinZ

1+m+rc

intm+rc


dx


c2scsumr=1

e(nr

c

) infin

intndashinfin


dx

Sincecsumr=1

e(nr

c

)=

c c | n0 c ∤ n

ζ(2s)1


infin

intndashinfin


dx


(42)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =

radic

πΓ(sndash 1

2)




Γ(s)infin

intndashinfin

e(ndashxy)(x2 + 1)s

dx =infin

int0

infin

intndashinfin


)sdx du

u

=infin

int0endashuus

infin

intndashinfin


=radic

πinfin


u


40 OWEN BARRETT




ress=1 E(z s) =3π



intΓH

∣∣ f (z)∣∣2 dx dyy2

lt infin




0 1)isin H








Am(y)e(mx)













intndashinfin

Iν(( 0 ndash1

1 0)middot( 1 u


=infin

intndashinfin

(y

(u + x)2 + y2

)νψ(ndashu) du

= ψ(x)infin

intndashinfin

(y

u2 + y2

)νψ(ndashu) du

42 OWEN BARRETT




W(z ν ψm) =radic

2(π|m|)νndash

12

Γ(ν)radic


where

Kν(y) =12

infin


u




where

W(y ν ψm) =infin

intndashinfin

(y

u2 + y2


= y1ndashνinfin

intndashinfin


du




Ψ(z) = aW(z ν ψ)





2(2π|m|y) andradic



Iν(y) =infinsumk=0

(12y)ν+2k

kΓ(k + ν + 1) and


sin πν=

12

infin


t

The asymptotics

Iν(y) simey

radic2πy


2y




f (z) = sumn=0

anradic




f (z) = sumnisinZ

An(y)e(nx)



44 OWEN BARRETT





form








Z =dcup

i=1

((gndash1Zg) cap Z

)δi

ZgZ =dcup

i=1Zgδi





by the formula

Tg( f (x)) =dsumi=1

f (gδix)



(45) ZgZ =cupi

Zαi ZhZ =cupj

Zβj

Then


ZgZβj =cupij

Zαiβj =cup

ZwsubZgZhZZw =

cupZwZsubZgZhZ

ZwZ



m(g hw)Tw

where



46 OWEN BARRETT


sumk

ckTgk



(n0n1 00 n0


by the matrices(

n0n1 00 n0





ad=n0le bltd

f(

az + bd

)




(( y x0 1))

= f(( y ndashx

0 1))




partsup2y

)



cupΔ





f(z) = sumn=0

a(n)radic





a(m)a(n) = a(mn) if (m n) = 1


a(mn

d2

)






48 OWEN BARRETT














N f sim2π(3radic



N f = N f + N f


measureradic






Then



and finally


radic3

2


1 ≪β intβ




t112ndashεf ≪ε



πt f



50 OWEN BARRETT


N fβ ≫ε t

112ndashεf




sumnge1


(1 ndash qn)24 = η(z)24 = Δ(z)





Fourier expansion

g(z) = sumngt0

ann(kndash1)2e(nz)


∣∣αp∣∣ = 1 =

∣∣∣βp


an le d(n)





∣∣ ∣∣∣βp

∣∣∣ le p764




52 OWEN BARRETT




N (R) = j radic

λj leX



(n2 + 1

)Rn



(4π)n2Γ(n2 + 1

)Rn



vol(partΩ)16π





R2 R rarr infin


(47) 0le rn leR

+

14π

R

intndashR

φprime

φ(12 + ir

)dr sim μ(F)

4πR2 R rarr infin







R2


54 OWEN BARRETT







∣∣∣2 dvol(z)







sumλj le λ

∣∣∣⟨h μj⟩∣∣∣2 ≪h

λlog λ




μt =∣∣∣∣E(z 1

2+ it)∣∣∣∣2 dvol(z)



limtrarrinfin

μt(A)μt(B)

=vol(A)vol(B)




56 OWEN BARRETT

3π c dx dy




∣∣∣φj





sumλj le λ

∣∣∣μj(ψ)∣∣∣2










Λd =

zQ =

ndashb +radic

d2a




(ndashb plusmn

radicd)





Λd cap ΩΛd



sumCisinΛd


sumCisinΛd









58 OWEN BARRETT

ψ isin Cinfin00(S

⋆1(X))

intΛd


ψ(u) dμL(u)




WEis(d t) =

1


2 + it)

d lt 01


(z 1

2 + it)

ds d gt 0

and

Wcusp(d t) =

1






2|d|ndash34Mu(d)

where

Mu(d) =

sumprimezisinΛ+

du(z) d lt 0











sumCisinΛd




)as |d| rarr infin






References






60 OWEN BARRETT













































62 OWEN BARRETT












1 Introduction










44 Weyls Law


46 Dukes Theorem

References

H 9THETRACEFORMULAbarrett/resources/obarrett... · 2017. 12. 24. · of any C2(Σg) function...

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Transcript of H 9THETRACEFORMULAbarrett/resources/obarrett... · 2017. 12. 24. · of any C2(Σg) function...