Maximum of the Riemann zeta function on a short interval of the critical line

Abstract

We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as $T\to \infty$ for a set of $t\in [T,2T]$ of measure $(1-o(1))T$, we have $$\underset{|t-u|\le 1}{\max }\log \left|\zeta \left(\frac{1}{2}+iu\right)\right|=(1+o(1))\log \log T.$$

arXiv comment28 pages, version accepted to Comm. Pure and Applied Math