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

TL;DR

This paper investigates the extreme values of the Riemann zeta function on random intervals of the critical line, demonstrating probabilistic bounds and confirming a weak version of a conjecture related to random matrix theory.

Contribution

It proves a probabilistic bound on the supremum of the zeta function's logarithm on random intervals, linking number theory with random matrix conjectures.

Findings

Supremum of real and imaginary parts of log zeta are within (1±ε) log log T with high probability.

Unconditional upper bound of log log T + g(T) for the supremum of Re log zeta.

Supports a weak version of Fyodorov, Hiary, and Keating's conjecture.

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.