Lower bounds for the large deviations of Selberg's central limit theorem

Abstract Let and . We prove that, for any and as , where is large enough depending on . 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 and Soundararajan and Radziwiłł and Soundararajan. The constant is explicit and is compared to the one conjectured by Keating and Snaith for the moments.