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

TL;DR

This paper derives an asymptotic formula for the mean values of the derivatives of Dirichlet L-functions at 1, providing insights into their distribution and bounds on large values for primitive characters.

Contribution

It introduces a new asymptotic formula for the 2k-th power mean of |L'/L(1, χ)| over primitive Dirichlet characters modulo prime q.

Findings

Asymptotic formula for mean values of |L'/L(1, χ)|

Bound on the number of characters with large |L'/L(1, χ)|

Implications for distribution of L-function derivatives

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.