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

Let $δ>0$ and $σ=\frac{1}{2}+\tfracδ{\log T}$. We prove that, for any $α>0$ and $V\sim α\log \log T$ as $T\to\infty$, $\frac{1}{T}\text{meas}\big\{t\in [T,2T]: \log|ζ(σ+\rm{i} τ)|>V\big\}\geq C_α(δ)\int_V^\infty \frac{e^{-y^2/\log\log T}}{\sqrt{π\log\log T}} \rm{d} y,$ where $δ$ is large enough depending on $α$. 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 $C_α(δ)$ is explicit and is compared to the one conjectured by Keating & Snaith for the moments.