An asymptotic formula for the $2k$-th power mean value of $\left| (L'/L)(1+it_0, \chi)\right|$

TL;DR

This paper derives an asymptotic formula for the average of the 2k-th power of the absolute value of the logarithmic derivative of Dirichlet L-functions at a fixed point, averaged over all non-principal characters modulo q.

Contribution

It provides a new asymptotic formula for the power mean values of the logarithmic derivatives of Dirichlet L-functions at a fixed point, extending previous results in analytic number theory.

Findings

Asymptotic formula for the 2k-th power mean value derived

Results hold for all non-principal characters modulo q

Applicable for fixed real t_0 and large q

Abstract

Let $q$ be a positive integer ($\geq 2$), $\chi$ be a Dirichlet character modulo $q$, $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, \chi)$ denote its derivative with respect to the complex variable $s$. Let $t_0$ be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $\left|(L^\prime/L)(1+it_0, \chi)\right|$ when $\chi$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_0=0$).