Modelling large values of L-functions

Transcription

1 Modelling large values of L-functions How big can things get? Christopher Hughes Exeter, 20 th January 2016 Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

2 How big can the Riemann zeta function get? ζ Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

3 How big can the Riemann zeta function get? ζ Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

4 How big can the Riemann zeta function get? ζ Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

5 Extreme values of the Riemann zeta function The running maxima of zeta for 0 t 10 3 Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

6 Large values of the Riemann zeta function The largest value of zeta over an interval of length 2π for t = 10 10, , , Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

7 Extreme values of zeta (Growth of maxima up to height T ) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

8 Extreme values of zeta Conjecture (Farmer, Gonek, Hughes) ( ( ) 1 ) max ζ( it) = exp 2 + o(1) log T log log T t [0,T ] Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

9 Bounds on extreme values of zeta Theorem (Littlewood; Ramachandra and Sankaranarayanan; Soundararajan; Chandee and Soundararajan) Under RH, there exists a C such that max ζ( it) = O t [0,T ] Theorem (Bondarenko-Seip) For all c < 1/ 2 max ζ( it) > exp t [0,T ] ( ( exp c ( C log T )) log log T ) log T log log log T log log T Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

10 An Euler-Hadamard hybrid Theorem (Gonek, Hughes, Keating) A simplified form of our theorem is: ζ( it) = P(t; X)Z (t; X) + errors where and P(t; X) = p X Z (t; X) = exp ( ( 1 1 p 1 2 +it ) 1 γ n Ci( t γ n log X) ) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

11 An Euler-Hadamard hybrid: Primes only Graph of P(t + t 0 ; X), with t 0 = γ , with X = log t 0 26 (red) and X = 1000 (green). Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

12 An Euler-Hadamard hybrid: Zeros only Graph of Z (t + t 0 ; X), with t 0 = γ , with X = log t 0 26 (red) and X = 1000 (green). Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

13 An Euler-Hadamard hybrid: Primes and zeros Graph of ζ( i(t + t 0)) (black) and P(t + t 0 ; X)Z (t + t 0 ; X), with t 0 = γ , with X = log t 0 26 (red) and X = 1000 (green). Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

14 Extreme values of zeta: RMT & zeros Keating and Snaith modelled the Riemann zeta function with Z UN (θ) := det(i N U N e iθ ) N = (1 e i(θn θ) ) n=1 where U N 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 = log T 2π Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

15 Extreme values of zeta: RMT & zeros Graph of the value distribution of log ζ( it) around the 1020 th zero (red), against the probability density of log Z UN (0) with N = 42 (green). Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

16 Extreme values of zeta: RMT & zeros Simply taking the largest value of a characteristic polynomial doesn t work. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

17 Extreme values of zeta: RMT & zeros Simply taking the largest value of a characteristic polynomial doesn t work. Split the interval [0, T ] up into M = T log T N blocks, each containing approximately N zeros. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

18 Extreme values of zeta: RMT & zeros Simply taking the largest value of a characteristic polynomial doesn t work. 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 N random unitary matrix. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

19 Extreme values of zeta: RMT & zeros Simply taking the largest value of a characteristic polynomial doesn t work. 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 N random unitary matrix. Find the smallest K = K (M, N) such that choosing M independent characteristic polynomials of size N, almost certainly none of them will be bigger than K. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

20 Extreme values of zeta: RMT & zeros Note that { P max max 1 j M θ } { Z (j) U (θ) K = P N max Z UN (θ) K θ } M Theorem Let 0 < β < 2. If M = exp(n β ), and if ( ) (1 K = exp 1 2 β + ε) log M log N then { P max max 1 j M θ as N for all ε > 0, but for no ε < 0. } Z (j) U (θ) K 1 N Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

21 Extreme values of zeta: RMT & zeros Recall ζ( it) = P(t; X)Z (t; X) + errors We showed that Z (t; X) can be modelled by characteristic polynomials of size N = log T e γ log X Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

22 Extreme values of zeta: RMT & zeros Recall ζ( it) = P(t; X)Z (t; X) + errors We showed that Z (t; X) can be modelled by characteristic polynomials of size N = log T e γ log X Therefore the previous theorem suggests Conjecture If X = log T, then ( ( 1 max Z (t; X) = exp 2 + o(1) ) ) log T log log T t [0,T ] Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

23 Extreme values of zeta: RMT & zeros Theorem By the PNT, if X = log T then for any t [0, T ], ( ( )) log T P(t; X) = O exp C log log T Thus one is led to the max values conjecture Conjecture ( ( ) 1 ) max ζ( it) = exp 2 + o(1) log T log log T t [0,T ] Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

24 Extreme values of zeta: Primes First note that P(t; X) = exp p X 1 p 1/2+it O(log X) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

25 Extreme values of zeta: Primes First note that P(t; X) = exp p X 1 p 1/2+it O(log X) Treat p it as independent random variables, U p, distributed uniformly on the unit circle. The distribution of Re U p p p X tends to Gaussian with mean 0 and variance 1 2 log log X as X. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

26 Extreme values of zeta: Primes We let X = exp( log T ) and model the maximum of P(t; X) by finding the maximum of the Gaussian random variable sampled T (log T ) 1/2 times. This suggests max P(t; X) = O t [0,T ] for all ε > 0 and no ε < 0. ( ( exp ( 1 + ε) )) log T log log T 2 Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

27 Extreme values of zeta: Primes We let X = exp( log T ) and model the maximum of P(t; X) by finding the maximum of the Gaussian random variable sampled T (log T ) 1/2 times. This suggests max P(t; X) = O t [0,T ] ( ( exp ( 1 + ε) )) log T log log T 2 for all ε > 0 and no ε < 0. For such a large X, random matrix theory suggests that ( ( )) max Z (t; X) = O exp log T. t [0,T ] This gives another justification of the large values conjecture. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

28 Large values of zeta (Distribution of maxima over short intervals) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

29 Distribution of the max of characteristic polynomials In 2012 Fyodorov, Hiary and Keating studied the distribution of the maximum of a characteristic polynomial of a random unitary matrix via freezing transitions in certain disordered landscapes with logarithmic correlations. This mixture of rigorous and heuristic calculation led to: Conjecture (Fyodorov, Hiary and Keating) For large N, log max Z UN (θ) log N 3 log log N + Y θ 4 where the random variable Y has the density 4e 2y K 0 (2e y ) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

30 Distribution of the max of characteristic polynomials The probability density of Y Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

31 Distribution of the max of characteristic polynomials Note that for large K. P {Y K } 2Ke 2K Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

32 Distribution of the max of characteristic polynomials Note that P {Y K } 2Ke 2K for large K. However, one can show that if K / log N but K N ɛ then { } P max log Z UN (θ) K θ = exp ( K 2 ) (1 + o(1)) log N Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

33 Distribution of the max of characteristic polynomials Note that P {Y K } 2Ke 2K for large K. However, one can show that if K / log N but K N ɛ then { } P max log Z UN (θ) K θ = exp ( K 2 ) (1 + o(1)) log N Thus there must be a critical K (of the order log N) where the probability that max θ Z U (θ) K changes from looking like linear exponential decay to quadratic exponential decay. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

34 Large values of zeta Their work with characteristic polynomials led Fyodorov and Keating to conjecture that ( ( ) T max ζ( 1 T t T +2π 2 + it) exp log log 3 ( ) ) T 2π 4 log log log + Y 2π with Y having (approximately) the same distribution as before. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

35 Large values of zeta Distribution of 2 log max t [T,T +2π] ζ( it) (after rescaling to get the empirical variance to agree) based on zeros near T = Graph by Ghaith Hiary, taken from Fyodorov-Keating. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

36 Large values of zeta The conjecture that for almost all T max log ζ( 1 0 h i(t + h)) = log log T 3 log log log T + O(1) 4 was backed up by a different argument of Harper, and later by Arguin, Belius and Harper. They considered random Euler products max 0 h 1 p T Re(U p p ih ) p where U p are uniform iid on the unit circle. Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

37 Large values of zeta Split the sum over primes up Y k (h) = 2 k 1 <log p 2 k Re(U p p ih ) p These are well-correlated if h h < 2 k and nearly uncorrelated for h and h further apart. Translating this as a branching random walk they were able to show that max 0 h 1 p T Re(U p p ih ) p = log log T 3 4 log log log T + o P(log log log T ) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

38 Moments at the local maxima Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

39 Moments of the local maxima Theorem (Conrey and Ghosh) As T 1 ζ( 1 N(T ) 2 + it n) 2 e2 5 log T 2 t n T where t n are the points of local maxima of ζ( it). This should be compared with Hardy and Littlewood s result 1 T T 0 ζ( it) 2 dt log T Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

40 Moments of the local maxima In 2012 Winn succeeding in proving a random matrix version of this result (in disguised form) Theorem (Winn) As N [ 1 E N N Z UN (φ n ) ] [ 2k C(k) E Z UN (0) 2k] n=1 where φ n are the points of local maxima of Z UN (θ), and where C(k) can be given explicitly as a combinatorial sum involving Pochhammer symbols on partitions. In particular, [ 1 E N N Z UN (φ n ) ] 2 n=1 e2 5 N 2 Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31

41 Bibliography Growth of zeta function D. Farmer, S. Gonek and C. Hughes The maximum size of L-functions Journal für die reine und angewandte Mathematik (2007) V. Chandee and K. Soundararajan Bounding ζ(1/2 + it) on the Riemann hypothesis Bull. London Math. Soc. (2011) A. Bondarenko and K. Seip Large GCD sums and extreme values of the Riemann zeta function (2015) arxiv: Euler-Hadamard product formula S. Gonek, C. Hughes and J. Keating, A hybrid Euler-Hadamard product for the Riemann zeta function Duke Math. J. (2007) Modelling zeta using characteristic polynomials J. Keating and N. Snaith Random matrix theory and ζ(1/2 + it) Comm. Math. Phys. (2000) Distribution of max values Y. Fyodorov, G. Hiary and J. Keating, Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function Phys. Rev. Lett. (2012) Y. Fyodorov and J. Keating, Freezing transitions and extreme values: random matrix theory, ζ(1/2 + it), and disordered landscapes (2012) arxiv: A. Harper, A note on the maximum of the Riemann zeta function, and log-correlated random variables (2013) arxiv: L.-P. Arguin, D. Belius and A. Harper, Maxima of a randomized Riemann Zeta function, and branching random walks (2015) arxiv: Moments at the local maxima J. Conrey and A. Ghosh, A mean value theorem for the Riemann zeta-function at its relative extrema on the critical line J. Lond. Math. Soc. (1985) B. Winn, Derivative moments for characteristic polynomials from the CUE Commun. Math. Phys. (2012) Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January / 31