The Fyodorov-Hiary-Keating Conjecture. I

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 $T\le t\le 2T$ for which $$\underset{|h|\le 1}{\max }|\zeta (1/2+it+ih)|>e^y\frac{\log T}{(\log \log T{)}^{3/4}}$$ is bounded by $Cy{e}^{-2y}$ uniformly in $y\ge 1$. 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.

Simplified Appendix B