Distribution of Dirichlet $L$-functions

TL;DR

This paper investigates the distribution of Dirichlet L-functions and their random models, analyzing differences across specific complex plane regions to deepen understanding of their value distributions.

Contribution

It provides a detailed comparison between the actual distributions of Dirichlet L-functions and their probabilistic models in different regions.

Findings

Distribution patterns differ between the regions $rac12< ext{Re} s<1$ and $ ext{Re} s=1$.

Quantifies the discrepancy between actual and modeled distributions.

Offers insights into the behavior of Dirichlet L-functions in critical regions.

Abstract

In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of $\frac12<\RE s<1$ and $\RE s=1$ separately.