The Fyodorov--Hiary--Keating Conjecture on Mesoscopic Intervals

Abstract

We derive precise upper bounds for the maximum of the Riemann zeta function on a typical short interval of the critical line. We show that for fixed $\theta\in(-1,0]$, large $T$, and $y\geq 2$ satisfying $y=O(\log\log T/\log\log\log T)$, the proportion of points $t\in [T,2T]$ for which \begin{align*} \max_{|h|\leq \log^\theta T}\big|\zeta(&\tfrac{1}{2}+it+ih)\big|>e^{y} \cdot e^{S\sqrt{(\log\log T)|\theta|/2}}\frac{(\log T)^{(1+\theta)}}{(\log\log T)^{3/4}} \end{align*} is bounded above by a constant times $y\exp({-2y-y^2/((1+\theta)\log\log T)})$, where $S=S(t)$ is a quantity whose value distribution is approximately that of a standard Gaussian. Up to a multiplicative constant, this settles the upper bound of a conjecture of Fyodorov--Hiary--Keating which was only known in the leading order for $\theta\in(-1,0)$. Using similar techniques, we also derive upper bounds for the second moment of the zeta function on such intervals. We show that for large $T$, the proportion of $t\in [T,2T]$ for which \begin{align*} \frac{1}{\log^\theta T}\int_{-\log^\theta T}^{\log^\theta T} \big|\zeta(&\tfrac{1}{2}+it+ih)\big|^2\mathrm{d}h > A e^{S\sqrt{2|\theta|\log\log T}} \frac{(\log T)^{(1+\theta)}}{\sqrt{\log\log T}} \end{align*} tends to zero as $A\to\infty$, for the same $S$ as above. 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 proofs use an adaptation of the recursive scheme introduced by one of the authors, Bourgade and Radziwi{\l}{\l}.