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 ≤ t ≤ 2T for which max_(|h|≤1|) ζ(1/2 + it +ih)| > e^y log T/((log log T)^(3/4)) is bounded by Cye^(−2y) uniformly in y ≥ 1. This is expected to be optimal for y = O(√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.