On the distribution of $\log |L(\sigma, \chi)|$ and $\log L(\sigma, \chi_D)$ in the modulus aspect

TL;DR

This paper investigates the distribution of logarithmic values of Dirichlet L-functions in different aspects, providing integral representations under GRH and constructing an unconditional density function for real characters.

Contribution

It offers new integral formulas for average values of log L-functions under GRH and constructs an unconditional M-function for real characters in the level aspect.

Findings

Distribution of $ ext{log}|L(\sigma,\chi)|$ can be expressed via integrals with an M-function.

Unconditional M-function constructed for $ ext{log}L(\sigma,\chi_D)$ in the D-aspect.

Results connect distributions in modulus and level aspects of L-functions.

Abstract

Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, \chi)|$ in $q$-aspect at a fixed $s=\sigma>1/2$, under Generalized Riemann Hypothesis (GRH), we explain it can be written as integrals involving the same density function ($M$-function) for the average of values of the difference between the logarithms of two symmetric power $L$-functions in the level aspect. For the distribution of values of $\log L(s, \chi_D)$ and $L'/L(s, \chi_D)$ in the $D$-aspect at a fixed $s=\sigma>1/2$ which $L(\sigma', \chi)\neq 0$ in $\sigma\leq \sigma' \leq 1$, where $\chi_D$ is a real character attached to a fundamental discriminant $D$, we construct a $M$-function unconditionally.