Large deviations in Selberg's central limit theorem

Abstract

Following Selberg it is known that as T → ∞, [formula] uniformly for Δ ≤ (log log log T)^((1/2) - ε). We extend the range of Δ to Δ « (log log T)^((1/10) - ε). We also speculate on the size of the largest Δ for which the above normal approximation can hold and on the correct approximation beyond this point