Selberg's central limit theorem of $L$-functions near the critical line

TL;DR

This paper extends Selberg's central limit theorem to a multi-dimensional setting for $L$-functions near the critical line, providing an asymptotic expansion for their distribution in a specific complex region.

Contribution

It introduces a multi-dimensional asymptotic expansion of Selberg's CLT for $L$-functions close to the critical line, generalizing previous one-dimensional results.

Findings

Derived an asymptotic expansion for the distribution of $L$-functions

Extended Selberg's CLT to multi-dimensional case near the critical line

Analyzed behavior for $ heta$ in (0, 1/2)

Abstract

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