An asymptotic distribution for $\mid L^{\mathrm{\prime }}\mathrm{/}L(1,\chi )\mid$

Let $\chi$ be a Dirichlet character modulo $q$, let $L(s,\chi )$ be the attached Dirichlet $L$-function, and let $L^{\mathrm{\prime }}(s,\chi )$ 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 $\mid L^{\mathrm{\prime }}\mathrm{/}L(1,\chi )\mid$ 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 $\mid L^{\mathrm{\prime }}\mathrm{/}L(1,\chi )\mid$ is large.