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

TL;DR

This paper proves the leading order of a conjecture regarding the maximum size of the Riemann zeta function on short intervals along the critical line, showing it grows roughly as log log T for most t in large intervals.

Contribution

It establishes the leading order of the conjecture by Fyodorov, Hiary, and Keating about the maximum of the zeta function on short intervals, a significant step in understanding its extreme values.

Findings

Maximum of log |zeta(1/2 + i u)| on short intervals is approximately log log T.

Holds for most t in [T, 2T] as T approaches infinity.

Confirms the conjectured growth rate of the zeta function's maximum.

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 $T \rightarrow \infty$ for a set of $t \in [T, 2T]$ of measure $(1 - o(1)) T$, we have $$ \max_{|t-u|\leq 1}\log\left|\zeta\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$