Large deviations of Selberg's central limit theorem on RH

Abstract: Assuming the Riemann hypothesis, we show that for $k>0$ and $V\sim k\log\log T$, \[ \frac{1}{T}\operatorname{meas}\bigg\{t\in[T,2T]: \log |\zeta(1/2+{\rm i} t)|>V\bigg\}\leq C_k \frac{e^{-V^2/\log\log T}}{\sqrt{\log\log T}}. \] This shows that Selberg's central limit theorem persists in the large deviation regime. As a corollary, we recover the result of Soundararajan and of Harper on the moments of $\zeta$. This directly implies the sharp moment bounds of Soundararajan and Harper, i.e., \[ \frac{1}{T}\int_T^{2T}|\zeta(1/2+{\rm i} t)|{\rm d}t\leq C_k (\log T)^{k^2}. \] This is joint work with Louis-Pierre Arguin (Oxford University) and Emma Bailey (University of Bristol).

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)