Modelling large values of L-functionsjulioandrade.weebly.com/uploads/4/0/3/2/40324815/... · Graph...

Modelling large values of L-functionsHow big can things get?

Christopher Hughes

Exeter, 20th January 2016

Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20th January 2016 1 / 31

How big can the Riemann zeta function get?

ζ


Extreme values of the Riemann zeta function

2000 4000 6000 8000

5

10

15

The running maxima of zetafor 0 ≤ t ≤ 103


Large values of the Riemann zeta function

0 1 2 3 4 5 6

5

10

15

20

25

30

0 1 2 3 4 5 6

5

10

15

20

25

30

0 1 2 3 4 5 6

5

10

15

20

25

30

0 1 2 3 4 5 6

5

10

15

20

25

30

The largest value of zeta over an interval of length 2πfor t = 1010,1010 + 100,1010 + 200,1010 + 300


Extreme values of zeta(Growth of maxima up to height T )


Extreme values of zeta

Conjecture (Farmer, Gonek, Hughes)

maxt∈[0,T ]

|ζ(12 + it)| = exp

(( 1√2+ o(1)

)√log T log log T

)


Bounds on extreme values of zeta

Theorem (Littlewood; Ramachandra and Sankaranarayanan;Soundararajan; Chandee and Soundararajan)Under RH, there exists a C such that

maxt∈[0,T ]

|ζ(12 + it)| = O

(exp

(C

log Tlog log T

))

Theorem (Bondarenko-Seip)

For all c < 1/√

2

maxt∈[0,T ]

|ζ(12 + it)| > exp

(c

√log T log log log T

log log T

)


An Euler-Hadamard hybrid

Theorem (Gonek, Hughes, Keating)A simplified form of our theorem is:

ζ(12 + it) = P(t ;X )Z (t ;X ) + errors

where

P(t ;X ) =∏p≤X

(1− 1

p12+it

)−1

and

Z (t ;X ) = exp

(∑γn

Ci(|t − γn| log X )

)


An Euler-Hadamard hybrid: Primes only

Graph of |P(t + t0;X )|, with t0 = γ1012+40,with X = log t0 ≈ 26 (red) and X = 1000 (green).


An Euler-Hadamard hybrid: Zeros only

Graph of |Z (t + t0;X )|, with t0 = γ1012+40,with X = log t0 ≈ 26 (red) and X = 1000 (green).


An Euler-Hadamard hybrid: Primes and zeros

Graph of |ζ( 12 + i(t + t0))| (black) and |P(t + t0;X )Z (t + t0;X )|,

with t0 = γ1012+40, with X = log t0 ≈ 26 (red) and X = 1000 (green).


Extreme values of zeta: RMT & zeros

Keating and Snaith modelled the Riemann zeta function with

ZUN (θ) := det(IN − UNe−iθ)

=N∏

n=1

(1− ei(θn−θ))

where UN is an N × N unitary matrix chosen with Haar measure.

The matrix size N is connected to the height up the critical line T via

N = logT2π



-10 -5 5

0.05

0.10

0.15

0.20

0.25

Graph of the value distribution of log |ζ( 12 + it)| around the 1020th zero (red),

against the probability density of log |ZUN (0)| with N = 42 (green).



Simply taking the largest value of a characteristic polynomial doesn’twork.

Split the interval [0,T ] up into

M =T log T

N

blocks, each containing approximately N zeros.Model each block with the characteristic polynomial of an N × Nrandom unitary matrix.Find the smallest K = K (M,N) such that choosing M independentcharacteristic polynomials of size N, almost certainly none of them willbe bigger than K .



Simply taking the largest value of a characteristic polynomial doesn’twork.Split the interval [0,T ] up into

M =T log T

N

blocks, each containing approximately N zeros.

Model each block with the characteristic polynomial of an N × Nrandom unitary matrix.Find the smallest K = K (M,N) such that choosing M independentcharacteristic polynomials of size N, almost certainly none of them willbe bigger than K .




M =T log T

N

blocks, each containing approximately N zeros.Model each block with the characteristic polynomial of an N × Nrandom unitary matrix.

Find the smallest K = K (M,N) such that choosing M independentcharacteristic polynomials of size N, almost certainly none of them willbe bigger than K .




M =T log T

N

blocks, each containing approximately N zeros.Model each block with the characteristic polynomial of an N × Nrandom unitary matrix.Find the smallest K = K (M,N) such that choosing M independentcharacteristic polynomials of size N, almost certainly none of them willbe bigger than K .



Note that

P{

max1≤j≤M

maxθ|Z

U(j)N(θ)| ≤ K

}= P

{maxθ|ZUN (θ)| ≤ K

}M

TheoremLet 0 < β < 2. If M = exp(Nβ), and if

K = exp(√(

1− 12β + ε

)log M log N

)then

P{

max1≤j≤M

maxθ|Z

U(j)N(θ)| ≤ K

}→ 1

as N →∞ for all ε > 0, but for no ε < 0.



Recallζ(1

2 + it) = P(t ;X )Z (t ;X ) + errors

We showed that Z (t ;X ) can be modelled by characteristic polynomialsof size

N =log T

eγ log X

Therefore the previous theorem suggests

ConjectureIf X = log T , then

maxt∈[0,T ]

|Z (t ;X )| = exp(( 1√

2+ o(1)

)√log T log log T

)



TheoremBy the PNT, if X = log T then for any t ∈ [0,T ],

P(t ;X ) = O

(exp

(C

√log T

log log T

))

Thus one is led to the max values conjecture

Conjecture

maxt∈[0,T ]

|ζ(12 + it)| = exp

(( 1√2+ o(1)

)√log T log log T

)


Extreme values of zeta: Primes

First note that

P(t ;X ) = exp

∑p≤X

1p1/2+it

×O(log X )

Treat p−it as independent random variables, Up, distributed uniformlyon the unit circle.The distribution of

Re∑p≤X

Up√p

tends to Gaussian with mean 0 and variance 12 log log X as X →∞.



We let X = exp(√

log T ) and model the maximum of P(t ;X ) by findingthe maximum of the Gaussian random variable sampled T (log T )1/2

times. This suggests

maxt∈[0,T ]

|P(t ;X )| = O(

exp((

1√2+ ε)

√log T log log T

))for all ε > 0 and no ε < 0.

For such a large X , random matrix theory suggests that

maxt∈[0,T ]

|Z (t ;X )| = O(

exp(√

log T))

.

This gives another justification of the large values conjecture.



We let X = exp(√

log T ) and model the maximum of P(t ;X ) by findingthe maximum of the Gaussian random variable sampled T (log T )1/2

times. This suggests

maxt∈[0,T ]

|P(t ;X )| = O(

exp((

1√2+ ε)

√log T log log T

))for all ε > 0 and no ε < 0.For such a large X , random matrix theory suggests that

maxt∈[0,T ]

|Z (t ;X )| = O(

exp(√

log T))

.

This gives another justification of the large values conjecture.


Large values of zeta(Distribution of maxima over short intervals)


Distribution of the max of characteristic polynomials

In 2012 Fyodorov, Hiary and Keating studied the distribution of themaximum of a characteristic polynomial of a random unitary matrix viafreezing transitions in certain disordered landscapes with logarithmiccorrelations. This mixture of rigorous and heuristic calculation led to:

Conjecture (Fyodorov, Hiary and Keating)For large N,

log maxθ|ZUN (θ)| ∼ log N − 3

4log log N + Y

where the random variable Y has the density 4e−2yK0(2e−y )



-2 2 4

0.1

0.2

0.3

0.4

The probability density of Y



Note thatP {Y ≥ K} ≈ 2Ke−2K

for large K .

However, one can show that if K/ log N →∞ but K � Nε then

P{

maxθ

log |ZUN (θ)| ≥ K}

= exp(− K 2

log N(1 + o(1))

)Thus there must be a critical K (of the order log N) where theprobability that maxθ |ZU(θ)| ≈ K changes from looking like linearexponential decay to quadratic exponential decay.




for large K .However, one can show that if K/ log N →∞ but K � Nε then

P{

maxθ


= exp(− K 2

log N(1 + o(1))

)

Thus there must be a critical K (of the order log N) where theprobability that maxθ |ZU(θ)| ≈ K changes from looking like linearexponential decay to quadratic exponential decay.




for large K .However, one can show that if K/ log N →∞ but K � Nε then

P{

maxθ


= exp(− K 2

log N(1 + o(1))

)Thus there must be a critical K (of the order log N) where theprobability that maxθ |ZU(θ)| ≈ K changes from looking like linearexponential decay to quadratic exponential decay.


Large values of zeta

Their work with characteristic polynomials led Fyodorov and Keating toconjecture that

maxT≤t≤T+2π

|ζ(12 + it)| ∼ exp

(log log

(T2π

)− 3

4log log log

(T2π

)+ Y

)with Y having (approximately) the same distribution as before.



Distribution of −2 log maxt∈[T ,T+2π] |ζ( 12 + it)| (after rescaling to get the

empirical variance to agree) based on 2.5× 108 zeros near T = 1028. Graphby Ghaith Hiary, taken from Fyodorov-Keating.



The conjecture that for almost all T

max0≤h≤1

log |ζ(12 + i(T + h))| = log log T − 3

4log log log T + O(1)

was backed up by a different argument of Harper, and later by Arguin,Belius and Harper.They considered random Euler products

max0≤h≤1

∑p≤T

Re(Upp−ih)√

p

where Up are uniform iid on the unit circle.



Split the sum over primes up

Yk (h) =∑

2k−1<log p≤2k

Re(Upp−ih)√

p

These are well-correlated if |h − h′| < 2−k and nearly uncorrelated forh and h′ further apart.Translating this as a branching random walk they were able to showthat

max0≤h≤1

∑p≤T

Re(Upp−ih)√

p= log log T − 3

4log log log T + oP(log log log T )


Moments at the local maxima


Moments of the local maxima

Theorem (Conrey and Ghosh)As T →∞

1N(T )

∑tn≤T

∣∣ζ(12 + itn)

∣∣2 ∼ e2 − 52

log T

where tn are the points of local maxima of |ζ(12 + it)|.

This should be compared with Hardy and Littlewood’s result

1T

∫ T

0|ζ(1

2 + it)|2dt ∼ log T


Moments of the local maxima

In 2012 Winn succeeding in proving a random matrix version of thisresult (in disguised form)

Theorem (Winn)As N →∞

E

[1N

N∑n=1

∣∣ZUN (φn)∣∣2k

]∼ C(k)E

[∣∣ZUN (0)∣∣2k]

where φn are the points of local maxima of∣∣ZUN (θ)

∣∣, and where C(k)can be given explicitly as a combinatorial sum involving Pochhammersymbols on partitions.

In particular,

E

[1N

N∑n=1

∣∣ZUN (φn)∣∣2] ∼ e2 − 5

2N


Bibliography

Growth of zeta function D. Farmer, S. Gonek and C. Hughes “The maximum size of L-functions” Journal für die reine undangewandte Mathematik (2007) 609 215–236

V. Chandee and K. Soundararajan “Bounding |ζ(1/2 + it)| on the Riemann hypothesis” Bull. LondonMath. Soc. (2011) 43 243–250

A. Bondarenko and K. Seip “Large GCD sums and extreme values of the Riemann zeta function” (2015)arXiv:1507.05840

Euler-Hadamard product formula S. Gonek, C. Hughes and J. Keating, “A hybrid Euler-Hadamard product for the Riemann zetafunction” Duke Math. J. (2007) 136 507–549

Modelling zeta using characteristic polynomials J. Keating and N. Snaith “Random matrix theory and ζ(1/2 + it)” Comm. Math.Phys. (2000) 214 57–89

Distribution of max values Y. Fyodorov, G. Hiary and J. Keating, “Freezing transition, characteristic polynomials of randommatrices, and the Riemann zeta-function” Phys. Rev. Lett. (2012) 108 170601

Y. Fyodorov and J. Keating, “Freezing transitions and extreme values: random matrix theory, ζ(1/2 + it),and disordered landscapes” (2012) arXiv:1211.6063

A. Harper, “A note on the maximum of the Riemann zeta function, and log-correlated random variables”(2013) arXiv:1304.0677

L.-P. Arguin, D. Belius and A. Harper, “Maxima of a randomized Riemann Zeta function, and branchingrandom walks” (2015) arXiv:1506.00629

Moments at the local maxima J. Conrey and A. Ghosh, “A mean value theorem for the Riemann zeta-function at its relativeextrema on the critical line” J. Lond. Math. Soc. (1985) 32 193–202

B. Winn, “Derivative moments for characteristic polynomials from the CUE” Commun. Math. Phys. (2012)315 531–563


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