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

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 $\sigma = \operatorname{Re}(s) > 1$.

We then discuss the main obstruction toward generalization to higher derivatives.