An asymptotic distribution for $\left|L^{\prime }/L(1,\chi )\right|$

Abstract

Let $\chi$ be a Dirichlet character modulo $q$, let $L(s,\chi )$ be the attached Dirichlet $L$-function, and let $L^{\prime }(s,\chi )$ denotes its derivative with respect to the complex variable $s$. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $\left|L^{\prime }/L(1,\chi )\right|$ when $\chi$ ranges a primitive Dirichlet character modulo $q$ for $q$ prime. We derive some consequences, in particular a bound for the number of $\chi$ such that $\left|L^{\prime }/L(1,\chi )\right|$ is large.

17 pages, 1 figure. Submitted to the Journal of the London Mathematical Society