Large deviations and continuity estimates for the derivative of a random model of log |ζ| on the critical line

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:472

Article

Arguin, Louis-Pierre^1,2  Ouimet, Frederic³ 

[1] CUNY, Baruch Coll, New York, NY 10010 USA

[2] CUNY, Grad Ctr, New York, NY 10010 USA

[3] Univ Montreal, Montreal, PQ H3T 1J4, Canada

关键词: Extreme value theory;    Large deviations;    Riemann zeta function;    Estimates;   

DOI  :  10.1016/j.jmaa.2018.11.044

来源: Elsevier

【 摘 要 】

In this paper, we study the random field X(h) (sic) Sigma(p <= T) Re(U-p p(-ih))/p(1/2), h is an element of [0,1], where (U-p, p primes) is an i.i.d. sequence of uniform random variables on the unit circle in C. Harper [7] showed that (X(h), h is an element of (0, 1)) is a good model for the large values of (log vertical bar zeta(1/2 + i(T + h))vertical bar, h is an element of [0,1]) when T is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin et al. [3] up to the second order, but the tightness of the recentered maximum is still an open problem. As a first step, we provide large deviation estimates and continuity estimates for the field's derivative X'(h). The main result shows that, with probability arbitrarily close to 1, max(h is an element of [0,1]) X(h) - max(h is an element of S) X(h) = O(1), where S a discrete set containing O(log T root log log T) points. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free