Lower bounds for the large deviations and moments of the Riemann zeta function on the critical line

TL;DR

This paper establishes new unconditional lower bounds for large deviations and moments of the Riemann zeta function on the critical line, matching known upper bounds and advancing understanding of its extreme value behavior.

Contribution

It provides the sharpest known unconditional lower bounds for fractional moments and large deviations of the Riemann zeta function on the critical line.

Findings

Unconditional lower bounds for large deviations of the zeta function.

Matching order of magnitude with known upper bounds for certain ranges.

Improved understanding of the zeta function's extreme value distribution.

Abstract

Building on work in \cite{AB24} on the Riemann zeta function at height $T$ off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order $V\sim\alpha\log\log T$ for any $\alpha>0.$ This gives another proof of the sharpest known unconditional lower bounds on the fractional moments of the Riemann zeta function, due to \cite{HSlower}. The lower bound on large deviations is of the same order of magnitude as the upper bound proved in \cite{AB23}, for the range $0<\alpha<2.$