Extreme Behaviour - How big can things get? - WarwickConjecture (Farmer, Gonek, Hughes) max t2[0;T]...

Extreme BehaviourHow big can things get?

Christopher Hughes

Venice, 9 May 2013

Christopher Hughes (University of York) Extreme Behaviour Venice, 9 May 2013 1 / 24

Motivation: The Riemann zeta function

The Riemann zeta function is of fundamental importance in numbertheory.

Understand the distribution of prime numbers.Prototypical L-function (Dirichlet, elliptic curve, modular form, etc).An interesting and challenging function to understand in its ownright.Universality (approximates any holomorphic function arbitrarilywell).Can be used as a “experimental” test-bed for certain physicalsystems.One of the most important open problems in modern mathematicsconcerns it.It’s popular!


The Riemann zeta function

2 4 6 8 10

-10

-5

5

10

A real version of the Riemann zeta function,plotted for 104 ≤ t ≤ 104 + 10


The Riemann zeta function

2 4 6 8 10

-10

-5

5

10

A real version of the Riemann zeta function,plotted for 1010 ≤ t ≤ 1010 + 10


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(

Clog T

log log T

))

Theorem (Montgomery; Balasubramanian and Ramachandra;Balasubramanian; Soundararajan)There exists a C′ such that

maxt∈[0,T ]

|ζ(12 + it)| = Ω

(exp

(C′√

log Tlog log T

))


Characteristic polynomials

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


Characteristic polynomials

-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).


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

)−1and

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).


RMT model for extreme values of zeta

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(12 + 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

)


Distribution of the max of characteristic polynomials

Fyodorov, Hiary and Keating have recently studied the distribution ofthe maximum of a characteristic polynomial of a random unitary matrixvia freezing transitions in certain disordered landscapes withlogarithmic correlations. This mixture of rigorous and heuristiccalculation led to:

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

log maxθ|ZUN (θ)| ∼ log N −

34

log log N + Y

where Y has the density P {Y ∈ dy} = 4e−2yK0(2e−y )dy



-2 2 4

0.1

0.2

0.3

0.4

The probability density of Y



This led them to conjecture 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.

This conjecture was backed up by a different argument of Harper,using random Euler products.



This led them to conjecture 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.This conjecture was backed up by a different argument of Harper,using random Euler products.



Distribution of −2 log maxt∈[T ,T+2π] |ζ( 12 + it)| (after rescaling to get theempirical variance to agree) based on 2.5× 108 zeros near T = 1028. Graph

by Ghaith Hiary, taken from Fyodorov-Keating.



Note thatP {Y ≥ K} ≈ 2Ke−2K

for large K .

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


Moments of the local maxima

Theorem (Conrey and Ghosh)As T →∞

1N(T )

∑tn≤T

∣∣ζ(12 + itn)∣∣2 ∼ e2 − 52 log Twhere tn are the points of local maxima of |ζ(12 + it)|.

This should be compared with Hardy and Littlewood’s result

1T

∫ T0|ζ(12 + it)|

2dt ∼ log T


Moments of the local maxima

Recently 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]∼ e

2 − 52

N


Summary

We looked at the rate of growth of the Riemann zeta function onthe critical line.The Riemann zeta function can be written in terms of a product ofits zeros times a product over all primes.The product over zeros can be modelled by ZUN (θ).One argument required knowing the large deviations ofmaxθ |ZUN (θ)|.The distribution of maxθ |ZUN (θ)| is now known.Its moderate deviations are still under study.The moments of the local maxima are not much bigger thanordinary moments.Extreme behaviour is rare!


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

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

Lower bound on growth rate K. Soundararajan “Extreme values of zeta and L-functions” Mathematische Annalen (2008) 342467–486

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

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

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

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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