Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function

TL;DR

Under the Riemann Hypothesis, the paper establishes upper bounds on large deviations and moments of the Riemann zeta function, refining previous results and employing a recursive proof scheme.

Contribution

It provides new conditional upper bounds for large deviations and moments of the zeta function, improving upon prior bounds with a novel recursive approach.

Findings

Bound on measure of large deviations of || under RH

Upper bounds for 2k-moments of ||

Recovery of Harper's moment bound

Abstract

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute constant $c>0$. This implies that the $2k$-moments of $|\zeta|$ are bounded above by $C_k(\log T)^{k^2}$, recovering the bound of Harper. The proof relies on the recursive scheme of one of the authors with Bourgade and Radziwill (2020), and combines ideas of Soundararajan (2009) and Harper (2013).