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

Abstract

Let $\delta >0$ and $\sigma =\frac{1}{2}+\frac{\delta }{\log T}$. We prove that, for any $\alpha >0$ and $V\sim \alpha \log \log T$ as $T\to \infty$, $\frac{1}{T}\text{meas}\{t\in [T,2T]:\log |\zeta (\sigma +\mathrm{i}\tau )|>\mathrm{V}\}\ge {\mathrm{C}}_{\alpha }(\delta ){\int }_{\mathrm{V}}^{\infty }\frac{{\mathrm{e}}^{-{\mathrm{y}}^2/\log \log \mathrm{T}}}{\sqrt{\pi \log \log \mathrm{T}}}\mathrm{d}\mathrm{y},$ 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 Radziwiłł & Soundararajan. The constant $C_{\alpha }(\delta )$ is explicit and is compared to the one conjectured by Keating & Snaith for the moments.