Distribution of values of higher derivatives of $L^{\prime }(s,\chi )/L(s,\chi )$

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 $\sigma =\mathrm{Re}(s)>1$. We then discuss the main obstruction toward generalization to higher derivatives.