An asymptotic expansion of Selberg's central limit theorem near the critical line

TL;DR

This paper derives an asymptotic expansion for the distribution of the Riemann zeta function near the critical line, refining Selberg's central limit theorem for specific complex values.

Contribution

It provides a new asymptotic expansion of Selberg's CLT for the zeta function close to the critical line at $rac{1}{2} + ( ext{log } T)^{- heta}$.

Findings

Asymptotic expansion of the zeta function distribution near the critical line.

Refinement of Selberg's central limit theorem for specific $ heta$ values.

Enhanced understanding of the zeta function's behavior in critical regions.

Abstract

We find an asymptotic expansion of Selberg's central limit theorem for the Riemann zeta function on $\sigma = \frac12 + ( \log T)^{-\theta}$ and $t \in [T, 2T]$, where $ 0 < \theta < \frac12$ is a constant.