An asymptotic formula for the $2k$-th power mean value of $\left|(L^{\prime }/L)(1+i{t}_0,\chi )\right|$

Abstract

Let $q$ be a positive integer ($\ge 2$), $\chi$ 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 $\left|(L^{\prime }/L)(1+i{t}_0,\chi )\right|$ when $\chi$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_0=0$).

35 pages, 3 figures