Distibution of values of higher derivatives of $L'(s,\chi)/L(s,\chi)$

TL;DR

This paper investigates the value distribution of higher derivatives of the logarithmic derivative of Dirichlet L-functions, extending previous results on M-functions for the first derivative to higher derivatives.

Contribution

It generalizes the value distribution theory of the first derivative of the logarithmic derivative of Dirichlet L-functions to higher derivatives.

Findings

Extended M-function results to higher derivatives

Analyzed the evolution of value distribution for derivatives beyond the first

Provided new insights into the behavior of derivatives of L-functions

Abstract

In this article we study the value distribution theory for the first derivative of the logarithmic derivative of Dirichlet $L$-functions, generalizing certain results of Ihara, Matsumoto et. al. related to ``$M$-functions" for $\sigma = $ Re$(s) > 1$. We then discuss how things evolve for higher derivatives.