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The Fyodorov-Hiary-Keating Conjecture. I.

Arguin, Louis-Pierre and Bourgade, Paul and Radziwiłł, Maksym (2020) The Fyodorov-Hiary-Keating Conjecture. I. . (Unpublished)

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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.

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Additional Information:The authors are grateful to Frederic Ouimet for several discussions, and to Erez Lapid and Ofer Zeitouni for their careful reading, pointing at a mistake in the initial proof of Lemma 23. The research of LPA was supported in part by NSF CAREER DMS-1653602. PB acknowledges the support of NSF grant DMS-1812114 and a Poincaré chair. MR acknowledges the support of NSF grant DMS-1902063 and a Sloan Fellowship.
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Alfred P. Sloan FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20210825-184537116
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110530
Deposited By: George Porter
Deposited On:26 Aug 2021 14:12
Last Modified:02 Jun 2023 01:08

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