The density function for the value-distribution of the Lerch zeta-function and its applications

TL;DR

This paper investigates the value-distribution of the Lerch zeta-function, establishing an explicit density function for its associated probability measure and deriving an asymptotic formula for zeros distribution.

Contribution

It introduces an explicit density function for the Lerch zeta-function's value distribution and connects it to the zeros count asymptotics, advancing understanding in analytic number theory.

Findings

Explicit density function for the Lerch zeta-function's value distribution

Asymptotic formula for zeros distribution related to the density

Enhanced understanding of the probabilistic behavior of zeta-functions

Abstract

The probabilistic study of the value-distributions of zeta-functions is one of the modern topics in analytic number theory. In this paper, we study a certain probability measure related to the value-distribution of the Lerch zeta-function. We prove that it has a density function which we can explicitly construct. Moreover, we prove an asymptotic formula for the number of zeros of the Lerch zeta-function on the right side of the critical line, whose main term is associated with the density function.