Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem

Abstract

Let $\delta>0$ and $\sigma=\frac{1}{2}+\tfrac{\delta}{\log T}$. We prove that, for any $\alpha>0$ and $V\sim \alpha\log \log T$ as $T\to\infty$, $\frac{1}{T}\text{meas}\big\{t\in [T,2T]: \log|\zeta(\sigma+\rm{i} \tau)|>V\big\}\geq C_\alpha(\delta)\int_V^\infty \frac{e^{-y^2/\log\log T}}{\sqrt{\pi\log\log T}} \rm{d} y,$ where $\delta$ is large enough depending on $\alpha$. The result is unconditional on the Riemann hypothesis. As a consequence, we recover the sharp lower bound for the moments on the critical line proved by Heap & Soundararajan and Radziwi{\l}{\l} & Soundararajan. The constant $C_\alpha(\delta)$ is explicit and is compared to the one conjectured by Keating & Snaith for the moments.