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

Abstract

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|ζ(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute constant $c>0$. This implies that the $2k$-moments of $|ζ|$ 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).