On certain mean values of logarithmic derivatives of $L$-functions and the related density functions

TL;DR

This paper constructs and analyzes density functions related to the value distribution of a broad class of $L$-functions, extending classical work on the Riemann zeta-function and connecting mean values to these densities.

Contribution

It generalizes the concept of density functions for $L$-functions beyond the Riemann zeta-function and links mean values to integrals involving these densities.

Findings

Density functions constructed for a wide class of $L$-functions.

Mean values of $L$-functions expressed as integrals with these densities.

Extension of classical density concepts to broader $L$-function families.

Abstract

We study some "density function" related to the value-distribution of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density function in a wide class of $L$-functions. We prove that certain mean values of $L$-functions in the class are represented as integrals involving the related density functions.