On the extreme values of the Riemann zeta function on random intervals of the critical line

Abstract

In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \epsilon > 0$, the supremum of the real and the imaginary parts of $\log \zeta (1/2 + it)$ for $t \in [UT -h, UT + h]$ are in the interval $[(1-\epsilon) \log \log T, (1+ \epsilon) \log \log T]$ with probability tending to $1$ when $T$ goes to infinity, if $U$ is uniformly distributed in $[0,1]$. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating, which has recently been intensively studied in the setting of random matrices. We also unconditionally show that the supremum of $\Re \log \zeta(1/2 + it)$ is at most $\log \log T + g(T)$ with probability tending to $1$, $g$ being any function tending to infinity at infinity.