An asymptotic formula for the $2k$-th power mean value of $|(L^{\prime}/L)(1+it_0, \chi)|$

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 $|(L^{\prime}/L)(1+it_0, \chi)|$ when $\chi$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_{0} = 0$).