Pacific Journal ofMathematics

A CLASS OF MODIFIED ζ AND L-FUNCTIONS

EMIL GROSSWALD AND F. J. SCHNITZER

Vol. 74, No. 2 June 1978

PACIFIC JOURNAL OF MATHEMATICS

Vol. 74, No. 2, 1978

A CLASS OF MODIFIED ζ AND L-FUNCTIONS

E. GROSSWALD AND F. J. SCHNITZER

The purpose of this paper is the construction of a classof functions that have exactly the same complex zeros asthe Riemann zeta function ζ(s), or as any Dirichlet functionL(s, χ). The motivation for this construction is found incertain attempts to study the Riemann hypothesis.

The problem of the Riemann hypothesis has been approached,occasionally

(a) by attempts to study functions that share with ζ(β)(s = σ+it)certain analytic properties (e.g., representation by Dirichlet series,functional equation, Euler product, etc.), in order to see what restric-tions these properties impose upon their zeros; or

(b) by the construction of functions, whose zeros are subject tocertain restrictions (e.g., they do, or they don't satisfy a "Riemannhypothesis"), in the hope to detect similarities with, or differencesfrom ζ(β), or L(s, χ).

To the first approach belong attempts to show that some of thoseanalytic properties suffice to impose some kind of "Riemann hypothesis."These attempts were not too successful, in part because it turnedout that functions like the the Epstein zeta function, with many ofthe properties of ζ(β) (functional equation, Dirichlet series-but noEuler product) may have zeros with σ > 1, and also with 0 < s < 1(see [4]).

To the second approach belongs, e.g., an attempt by Rademacher[5] to disprove the Riemann hypothesis, by studying the class offunctions for which, assuming "the Riemann hypothesis," the sumX r 7"1 sin Ίt — f{t) (7 = imaginary part of the complex zero 1/2 + iΎ)exhibits the discontinuities known to occur when p = lf2 + iΎ are zerosof ζ(s). It was shown, however, by Rubel and Straus (see [6] and[7]) that the known behaviour of f(t) for ζ(s) is implied alreadyby conditions much weaker than the Riemann hypothesis.

The results of the present paper seem to indicate that the lastapproach is unlikely to lead to interesting conclusions, but suggest anew and potentially useful approach. Indeed, we construct a class offunctions of analytic character very different from that of ζ(s) and,nevertheless, with the same complex zeros. We also sketch theconstrution of functions that share their complex zeros with Dirichlet'sL-functions. In principle, the construction is valid for all Dirichletseries with an Euler product. These new functions have a Dirichlet

357

358 E. GROSSWALD AND F. J. SCHNITZER

series representation, also an Euler product, but, in general, theydo not satisfy any functional equation and often cannot even becontinued analytically beyond σ > 0.

Section 2 contains the principle of the construction and the state-ments of the main results. Sections 3 and 4 contain the proofs,Section 5 discusses analytic continuations and Section 6 mentionssome possible applications.

2* Main results* Let pn be the nth prime and select qn so that

( 1 ) Pn^Qn^ Pn+l

With these qn form the infinite product

(2) ζ*(β) = Π (1 - ϊ ί )"1

The product converges absolutely for σ > 1 and uniformly σ ^ 1 + ε(ε > 0), so that ζ*(s) is holomorphic for σ > 1.

THEOREM 1. The function ζ*(s) possesses the following properties:(i) ζ*(s) Φ 0 for σ > 1; (ii) ζ*(s) can be continued as a meromorphicfunction in σ > 0; (iii) in σ > 0, ζ*(s) has a single pole at 8 = 1with residue r, 1/2 ^ r ^ 1; (iv) in σ > 0, ζ*(s) /ms £fee same zeros,with the same multiplicities as ζ(s).

Theorem 1 remains valid if qn is restricted only by pn ^ qn forall n ^ wo> with lim^oo pjqn = 1. In general, any restriction on <jrΛmay be applied only to subscripts n ^ n0 (n0 arbitrarily large). Manyother generalizations of Theorem 1 are also possible.

THEOREM 2. Let L(s, χ) = ΣϊU λ O ) ^ = Π* (1 - X(v)v~T\ whereχ(n) is a Dirichlet character modulo the natural integer k. ForK^h, select rational integers qn that satisfy the two conditions

(3) pn ^ qn ^ pn + K and pn = qn (mod k) .

For σ > 1 define L*(s, χ) by the absolutely convergent infinite product

(4) L*(s, χ) = π a - χ(<7)<rr,

extended over all q = qn. The function L*(s, χ) can be continuedinto the whole half plane σ > 0, where it has exactly the same zeros(including multiplicities) as L(s, χ). If χ(n) is not the principalcharacter, then L*(s, χ) is holomorphic in σ > 0.

The sketch of the proof of Theorem 2 (see §4) will suggest tothe reader a number of more general possible versions of the theorem.

A CLASS OF MODIFIED ζ AND L-FUNCTIONS 359

3. Proof of Theorem 1* Set

(5) φ{s) = Π id - JΓ')(1 - β-r'} ,P

where p runs through the primes pn and q through the correspondingreal numbers qn; here and in what follows the subscript n is suppressedwhenever this does not lead to ambiguities. For σ > 1, the infiniteproduct (5) converges absolutely (in fact, each factor convergesseparately) and

( 6 ) ζ*(β) = <p(s)ζ(s) .

LEMMA 1. The infinite product (5) converges absolutely inσ > 0, where φ(s) Φ 0. The convergence is uniform on compact setsσ0 <* σ ^ σlf \ 11 ^ T, for any given constants with σx > σ0 > 0 andT>0.

Assuming Lemma 1, (6) may be used to define the analytic con-tinuation of ζ*(s) to the whole set into which φ(s) can be continued,which contains the half plane σ > 0. From this all statements ofTheorem 1 follow immediately, except the value of the residue r.For real s one obtains, by using the uniformity of the convergence,that

r = lim {ζ*(s)/ζ(s)} - Km Π {1(1 - P"s)(l - g^DS-+1+ β-»l+ p

= Π {(1 - P-'XI - Q'T1} = lim Π id - JΓ'XIp-

π {(i - ίCXi - q T1}N-*oo n=l

By (1) and p, = 2 it follows that

1 = 1 - 1 = lim ί ί {(1 - iCXl - PίiiΓ1} ^ r2 2 -zv-oo »=i

^ lim fi {1 - 3>Ϊ1X1 - V«rι) = 1 ,N-*oo n=l

as claimed.The proof of Theorem 1 has been reduced to that of Lemma 1.

In its proof (and also in that of Lemma 3) we shall need several,presumably wellknown inequalities, which we collect, for convenience,in

LEMMA 2. For integral, rational m0 and real cc, xf y, ε, and u,with 0 < α ? < l , 0 < ? / < l , 0 < ε < 1/2, u > 0, the following inequalitieshold:

360 E. GROSSWALD AND F. J. SCHNITZER

( i ) 1 - cos a < α2/2;(ii) |loga;|<(l-flOAc;(iii) 1 < (1 - ί»)/(l - xy) < y-\ with l i π w - {(1 - x)/(l - xy)} = y~ι;(iv) if f(x) = Σ S - ,+1 »»~ V", then 0 < f(x) < x™°+ίj(l - x)(m0 + 1 ) ;

( v ) 1 - (1 - e) < (1 + e)(l - β—).

Proof of Lemma 2. (i) is classical, (ii) follows from

I log x\ = |log(l - (1 - aθ)| = Σ » - 1 ( l ~ *)" < (1 - *) Σ (1 - &)"% = 1 % = 0

= ( 1 - a?)/a? .

For (iii), observe that x < xy <1 and the first inequality holds. Setg(xf y) = 1 + #2/ — ̂ — #*; the second inequality is equivalent tog(x, »)>0. This follows from #(1, y) = 0 and {dg/dx}a<1 = y(l - α ^ X O .The limit for α?—>1~ may be obtained by LΉospitaΓs rule. Toprove (iv), consider f'(x) = Σm=m0+i %m+1 = #w°/(l - a?), so that0 < f{x) < [ V ° ( l - w ) " 1 ^ < (1 - a?)"1 ["u^du = xwo+1/(l - a?)(m0 + 1).

Jo Jo

For (v), set h(u) = (1 + e)(l — e~εu); then (v) is equivalent to h(u) > 1,for u > 0. We verify that Λ(0) = 1 and h'(u) > 0 for % > 0. Thefirst is obvious and the second is equivalent to (e~ε/(l~e))u(l + e) >—e'Mogίl — ε). As β~ε > 1 — ε, we only need to have 1 + ε >Σ?=i sw"Vw, or 1/2 > Σ^U ε71"2/^, easily verified to hold, say, forε < 3/5.

Proof of Lemma 1. We consider

log φ) = Σ {log (1 - P~s) - log(l - <ΓS)}

where the interchange of summations is easily justified for σ > 1.By the absolute and uniform convergence of the double series forσ *> 1 + ε, it is sufficient to study its convergence on compact setsC of the form σQ ^ σ <; 1, 11 \ ̂ T, with given σ0 > 0 and T > 0. Forfixed s = a + ίέ 6 C, define m0 = [cr1] + 1, where [x] stands for thegreatest integer function. We split the sum over m into two parts,Σϊ=i = Σm°=i = Σm2=Wo+] + Σ 1 + Σ 2 , say. The absolute and uniform con-vergence of Σ 2 is almost obvious. Indeed,

Σ Σ m~\q-ms - p~m=mo+l p

^ Σ m,-1 Σ (q~m° + p~m°)

m=m o +l

™>~ι Σ Vmσ = 2 Σ Σl

Σm>m 0

A CLASS OF MODIFIED ζ AND L-FUNCTIONS 361

By Lemma 2(iv) with x — p~% it follows that the inner sum is majorized

by

(1 - p~°)(m0 + 1) = (1 - 2-σ°)(m0 + l)pσm°+σ p1+σ*'

with ex = (1 - 2-σή-\ because σm0 + σ = tffltf"1] + 2) > cr^"1 + 1) =1 + o ̂ 1 + σQ. Here and in what follows, cif c2, stand for constantsdepending at most on σ0 and T.

It now follows that I Σ Ί ^ 2^PP~{1+ao) and the right-hand sideconverges and does not depend on s.

As for Σ 1 , the outer sum contains the finitely many terms with1 <̂ m 5̂ m0, and m0 = [σ"1] + 1 <̂ tf"1 + 1 < σ^1 + 1. Hence, it issufficient to prove the absolute and uniform convergence of the innersums ΣIP \v~ms ~ Q~ms\ for 1 <: m ̂ m0. Let qn = pn + r ^ p w + j ; then(see, e.g., [2] (14.3), page 131) r = r(p) ^ p 3 / 5 + ε for every ε > 0,provided that p Ξ> P(e). I t follows that 0 ^ r(p) ^ p4/δ, except,perhaps for finitely many primes, which have no influence upon theconvergence of the series. If x = x{p) = p/q, then 1 — r/q — (q — r)/q =p/q = a.(p) g 1, and, for p ̂ P(l/5), aj(p) > 1 - q~lf\ so that lim^oo x(p) =1". In particular, for sufficiently large p, a?(p)2 > 1/3. It followsthat \p~~ms - q~ms\ = p~ m σ | l — £ m s | = p~ m σ | l — ̂ v ί m l o g x | . If p = q,this difference vanishes; otherwise, a? < 1, r > 0, and the last factorcan be estimated for 3~1/2 <^ x <1 and m < m0, by using (i), (ii), and(iii) of Lemma 2, with 0 < y = mσ < 1, as follows:

11 - x™>eitml°** |2 = (1 - ^m σ cos(£m log x))2 + ^2 w σ sin2 (tm log α?)

= (1 - xmσf + 2^mσ(l - cos(ίm log a?)) < (1 - xmσf

+ xmσ(tm log xf ^ (1 - #mσ)2 + xma

1 - α;m

X ) V l ( 1 - x m σ ) 2 { l + (σm)-2x~2t2m2}— xm

- (1 - xmσ)\l + έVα V2)

^ (1 - xmσ)\l + 3Γ2/σ2) ,

so that |1 - x™e

itml08x\ < cί(l - xwσ), with cj = (1 + 3T 2 K) 1 / 2 . Thissequence of inequalities does not hold for the last term of the sum,with m = m0, because for it mσ = moσ = ([σ"1] + 1) > o~ισ = 1. Inthat case, however, (1 — x)/(l — xmQσ) < 1, so that

= (1 - α;wσ)2(l + x-'tXiσ-1] + I)2) ^ (1 - ^w σ)2(l + x^tXσo1 + I)2)

+ I)2)

362 E. GROSSWALD AND F. J. SCHNITZER

and

with c2 = (1 + δ FOo-1 +1) 2) 1 / 2 As c2 > c2, the inequality 11 - ^ V ί m l o g a Ί <c2(l — #mσ) holds for all m with 1 ^ m <: m0.

If the inner sums are extended only over the primes p < pN,the error of each sum is majorized by

Σ I P ~ W 8 - Q~ms\ = Σ

a telescoping series, with sum c2pχmσ. It follows that the total errorof Σ 1 is majorized by c2 Σ ϊ - i PNmσ° ̂ c2m0pπσ° and can be made arbi-trarily small, by taking N sufficiently large. This estimate is inde-pendent of s and the uniform convergence of the double series Σ S °fthe absolute values m~ί\q~ms—p~m8\ is proved. From the convergenceof the series for log φ{s) it also follows that φ(s) Φ 0 and this finishesthe proof of Lemma 1, hence also that of Theorem 1.

4* Proof of Theorem 2. Let

(7) φχ(8) = π {(i - χ(p)p-s)(i - χ(Q)Q-T1}.p

Then

( 8 ) L*(s, χ) = φχ(s)L(s, χ)

and Theorem 2 is an immediate corollary of

LEMMA 3. For σ>0, the infinite product (7) converges absolutelyand φχ(s) Φ 0. The convergence is uniform on compact sets σQ ^a <i σlf 111 ̂ T, for any constants σ0, σί9 T, that satisfy σ} > σ0 >0, T > 0.

Proof of Lemma 3. The absolute convergence of (7) for σ > 1is obvious; hence, it is sufficient to consider the convergence on thestated compact sets. As in the proof of Lemma 1, we take the loga-rithm of cpx(s) and, by taking into account also (3) and the absoluteconvergence for σ > 1, we get

logφx (β) = ΣΣm- ι {χ((Γ)?-" - X(pm)p-ms}(9)

m—lχ(Pm)(q~ms - P'MS)

A CLASS OF MODIFIED ζ AND L-FUNCTIONS 363

Hence, \logφχ(s)\ is majorized by Σm=iW~1Σ3)|^~ms—ί>~ms|. The uniformconvergence of this series is proved exactly as in the proof of Lemma1, except for minor details in the handling of Σ 1 . Its outer sum,we recall, contains no more than m0 ^ σ^1 + 1 terms; this bound doesnot depend on s, so that it is sufficient to prove the uniform con-vergence of the inner sums. The terms with q = p vanish. In theothers, set x = p/q and observe that 0 < a? < 1. lΐ q = p -{- r andε = r/q, then r ^ K, s ^ K/q and 1 - K/p< 1 - K/q ^ 1 - ε = 1 - r/q =(q — r)/q = P/Q = & < 1. Except for finitely many primes, ε = r/qstays below any preassigned positive quantity, so that l i m ^ x(p) =1". It follows as before that

|1 - x

mσeitmlogx\ < c2(l - xmσ) = c 2 (l-(l - ε)mσ) .

By Lemma 2(v), this is less than c8(l — e~εmσ) < cBεmσ <̂ czmσK/q <Zc4mσ/p ̂ cjp. Here c3 <; (1 + ε)c2 ^ (3/2)c2, c4 = Kc3 and c5 may betaken equal to 2c4, because ma ^ moσ ^ (σ"1 + l)σ = 1 + a <Ξ 2. Itnow follows that

\χ(qm)q~ms - %(pm)p~ms\ ^ p~mσ\l - χ™eitmlosx

By summing only over p < pN, each sum has an error not in excessof ΣAP>PN CδP"1"00 < V> w i t h - V arbitrarily small, provided that pN =pN(σOf T, η) is chosen large enough. This choice does not depend on8 and leads to a total error on Σ 1 that is majorized by mj] < (071 +1)57.The proof of the absolute and uniform convergence of (9) is complete,and with it the proofs of Lemma 3 and of Theorem 2.

5* Analytic continuation* We have shown that φ(s) and, moregenerally, φx(s) can be continued analytically into the whole halfplane σ > 0. By (6) and (8) this implies the analytic continuabilityto σ > 0 for ζ*(s) and L*(s, χ). In some cases it is obvious thatφ{s), or even φx(s) can be continued as meromorphic functions intothe whole complex plane, e.g., if qn = pn+ί. On the other hand,one may select the qn's so that σ — 0 becomes a natural boundaryfor φ(s) (see [1] for a similar result). Indeed, let all qn Φ pn, pn+1;then φ(s) has poles at all the points s = it, with t = tn>k =2kπ(\og qn)"\ keZ. One may select the real numbers qn so that thepoles itn>k will be dense on the imaginary axis. Apparently, thiswill be the case, whenever at most a finite number of factors 1 — q~8

in the denominator of φ(s) are cancelled by identical factors in thenumerator, but we have not pursued this matter further. Similarconsiderations hold for φx(s).

When σ = 0 is a natural boundary for φ(s)9 then it also is one

364 E. GROSSWALD AND F. J. SCHNITZER

for ζ*(s); hence, ζ(s) and ζ*(s), while sharing all their complex zeros,have an entirely different analytic character and a similar statementholds for L(s, χ) and L*(s, χ).

6* Possible applications* In 1948 Turan ([8]; see also [9]) showedthat if the partial sums ζn(s) = Σϊ=i m~s of the ζ-function do notvanish in the half planes σ > 1 + wr1/2+ε(ε > 0), then the Riemannhypothesis holds. Recently [3] H. Montgomery showed, however,that these partial sums do, in fact, vanish for sufficiently large neven in the half planes 1 + c(log log n)(\og n)~\ provided that c <(4 — π)/π, that bound being best possible. The question arises, whetherit is possible to choose the qn so that Turan's theorem remains valid,while Montgomery's construction may not apply.

Another possible approach is the following: The functions holo-morphic in the half-plane σ > 0 form a ring H under ordinary additionand multiplication. Within H the functions with the same zeros inσ > 0 as (s — l)ζ(s) form an ideal I, to which belong all functions(s — l)ζ*(β). The study of I and of H/I may throw some light onthe problem of the Riemann hypothesis. We intend to return to thistopic on a later date.

REFERENCES

1. E. Landau und A. Walfisz, Uber die Nichtfortsetzbarkeίt einiger durch DirichletscheReihen definierter Funktionen, Rendiconti di Palermo, 44 (1920), 82-86.2. H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes inMathematics No. 227. Springer-Verlag, Berlin, Heidelberg, New York, 1971.3# 1 Zeros of approximations to the zeta-function, Manuscript to be published.4. H. S. A. Potter and E. C. Titchmarsh, The zeros of Epstein's zeta-function, Proc.London Math. Soc. (2), 39 (1935), 372-384.5. H. Rademacher, Remarks concerning the Riemann-von Mangoldt formula, Reportof the Institute in the Theory of Numbers, University of Colorado, Boulder (Colorado)(1959), 31-37.6. L. A. Rubel and E. G. Straus, Special trigonometric series and the Riemann-vonMangoldt formula, Proc. of the 1963 Number Theory Conference, Univ. of Colorado,Boulder (Colorado) (1963), 69-70.7. , Special trigonometric series and the Riemann Hypothesis, Math. Scand.,18 (1966), 35-44.8. P. Turin, On some approximative Dirichlet-polynomials in the theory of the zetafunction of Riemann, Danske Vid. Selsk. Mat.-Fys. Medd., 24 (1948), No. 17, 36 pp.9. , Nachtrag zu meiner Abhandlung "On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann, Acta Math. Acad. Sci.Hungar., 10 (1959), 277-298.

Received July 12, 1977 and in revised form August 22, 1977.

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Pacific Journal of MathematicsVol. 74, No. 2 June, 1978

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ring injects into the K -theory of its field of quotients . . . . . . . . . . . . . . . . . . . . . . . 497Mitchell Herbert Taibleson, The failure of even conjugate characterizations of H1

on local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Keti Tenenblat, On characteristic hypersurfaces of submanifolds in Euclidean

space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507Jeffrey L. Tollefson, Involutions of Seifert fiber spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 519Joel Larry Weiner, An inequality involving the length, curvature, and torsions of a

curve in Euclidean n-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531Neyamat Zaheer, On generalized polars of the product of abstract homogeneous

polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

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