Distribution of the values of the derivative of the Dirichlet $L$-functions at its $a$-points

Abstract

In this paper, we study the value distribution of the derivative of a Dirichlet $L$-function $L'(s,\chi)$ at the $a$-points $\rho_{a,\chi}=\beta_{a,\chi}+i\gamma_{a,\chi}$ of $L(s,\chi).$ We give an asymptotic formula for the sum $$\sum_{\rho_{a,\chi};\ 0<\gamma_{a,\chi}\leq T}L'\left(\rho_{a,\chi},\chi\right) X^{\rho_{a,\chi}}\ \ \hbox{as}\ \ T\longrightarrow \infty,$$ where $X$ is a fixed positive number and $\chi$ is a primitive character $\mod q$. This work continues the investigations of Fujii \cite{2,3,4}, Garunk$\check{s}$tis \& Steuding \cite{7} and the authors \cite{12}.