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

Let $χ$ be a Dirichlet character modulo $q$, let $L(s, χ)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, χ)$ 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 $\left|L^\prime/L(1, χ)\right|$ when $χ$ ranges a primitive Dirichlet character modulo $q$ for $q$ prime. We derive some consequences, in particular a bound for the number of $χ$ such that $\left|L^\prime/L(1, χ)\right|$ is large.

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