Arguin, Louis-Pierre and Bourgade, Paul and Radziwiłł, Maksym (2020) The Fyodorov-Hiary-Keating Conjecture. I. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210825-184537116
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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.
Item Type: | Report or Paper (Discussion Paper) | ||||||||||||
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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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DOI: | 10.48550/arXiv.2007.00988 | ||||||||||||
Record Number: | CaltechAUTHORS:20210825-184537116 | ||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20210825-184537116 | ||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
ID Code: | 110530 | ||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||
Deposited By: | George Porter | ||||||||||||
Deposited On: | 26 Aug 2021 14:12 | ||||||||||||
Last Modified: | 02 Jun 2023 01:08 |
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