The Fyodorov-Hiary-Keating Conjecture on Mesoscopic Intervals

TL;DR

This paper proves precise upper bounds for the maximum of the Riemann zeta function on short, mesoscopic intervals, confirming a strong form of the Fyodorov-Hiary-Keating conjecture and extending previous results.

Contribution

It establishes the first strong, precise bounds for zeta maxima on mesoscopic intervals, advancing understanding of its extreme value distribution.

Findings

Bound the probability of large zeta maxima on short intervals.

Derive upper bounds for the second moment of zeta on mesoscopic scales.

Generalize previous results to a broader class of intervals.

Abstract

We derive precise upper bounds for the maximum of the Riemann zeta function on short intervals on the critical line, showing for any $\theta\in(-1,0]$, the set of $t\in [T,2T]$ for which $$\max_{|h|\leq \log^\theta T}|\zeta(\tfrac{1}{2}+it+ih)|>\exp\bigg({y+\sqrt{(\log\log T)|\theta|/2}\cdot {S}}\bigg)\frac{(\log T)^{(1+\theta)}}{(\log\log T)^{3/4}}$$ is bounded above by $Cy\exp({-2y-y^2/((1+\theta)\log\log T)})$ (where $S$ is a random variable that is approximately a standard Gaussian as $T$ tends to infinity). This settles a strong form of a conjecture of Fyodorov--Hiary--Keating in mesoscopic intervals which was only known in the leading order. Using similar techniques, we also derive upper bounds for the second moment of the zeta function on such intervals. Conditioning on the value of $S$, we show that for all $t\in[T,2T]$ outside a set of order $o(T)$, $$\frac{1}{\log^\theta T}\int_{|h|\in \log^\theta T} |\zeta(\tfrac{1}{2}+it+ih)|^2\mathrm{d}h \ll e^{2S}\cdot \left(\frac{(\log T)^{(1+\theta)}}{\sqrt{\log\log T}}\right).$$ This proves a weak form of another conjecture of Fyodorov-Keating and generalizes a result of Harper, which is recovered at $\theta = 0$ (in which case $S$ is defined to be zero). Our main tool is an adaptation of the recursive scheme introduced by one of the authors, Bourgade and Radziwi{\l}{\l} to mesoscopic intervals.