Large Deviation Estimates of Selberg's Central Limit Theorem and Applications

Abstract

For $V\sim \alpha \log\log T$ with $0<\alpha<2$, we prove \[ \frac{1}{T}\text{meas}\{t\in [T,2T]: \log|\zeta(1/2+ {\rm i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This improves prior results of Soundararajan and of Harper on the large deviations of Selberg's Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwi{\l}{\l} and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $(\log T)^\theta$, $0<\theta <3$, that is expected to be sharp for $\theta > 0$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwi{\l}{\l} and one of the authors to prove fine asymptotics for the maximum on intervals of length $1$.