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

Louis-Pierre Arguin, David Belius, Paul Bourgade, Maksym Radziwiłł, Kannan Soundararajan

Published 2016-12-27Version 1

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, if $t$ is uniformly distributed in $[T,2T]$, then $$ \max_{|t-u|\leq 1}\log\left|\zeta\left(\frac{1}{2}+\mathrm{i} u\right)\right|=(1+{\mathrm o}(1))\log\log T $$ with probability converging to 1 as $T\to\infty$.