The Fyodorov-Hiary-Keating Conjecture. I

TL;DR

This paper proves a strong upper bound on the measure of points where the Riemann zeta function's maximum exceeds a certain threshold in short intervals, advancing understanding of its extreme value distribution.

Contribution

It establishes a sharp, uniform upper bound for the maximum of the zeta function in short intervals, confirming part of the Fyodorov-Hiary-Keating conjecture.

Findings

Bound on measure of large zeta maxima in short intervals

Uniform decay rates in the parameter y

Sharper results than those known for random matrices

Abstract

By analogy with conjectures for random matrices, Fyodorov-Hiary-Keating and Fyodorov-Keating proposed precise asymptotics for the maximum of the Riemann zeta function in a typical short interval on the critical line. In this paper, we settle the upper bound part of their conjecture in a strong form. More precisely, we show that the measure of those $T \leq t \leq 2T$ for which $$ \max_{|h| \leq 1} |\zeta(1/2 + i t + i h)| > e^y \frac{\log T }{(\log\log T)^{3/4}}$$ is bounded by $Cy e^{-2y}$ uniformly in $y \geq 1$. This is expected to be optimal for $y= O(\sqrt{\log\log T})$. This upper bound is sharper than what is known in the context of random matrices, since it gives (uniform) decay rates in $y$. In a subsequent paper we will obtain matching lower bounds.