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

Let δ>0 and σ=1/2+δ/log T. We prove that, for any α>0 and V~ αlog log T as T→∞, 1/Tmeas∈ [T,2T]: log|ζ(σ+i τ)|>V\≥ C_α(δ)∫V^∞ e-y²/loglog T√πloglog T 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 Radziwi{}{} & Soundararajan. The constant C_α(δ) is explicit and is compared to the one conjectured by Keating & Snaith for the moments.