Distribution of the values of the derivative of the Dirichlet $L$-functions at its $a$-points
Bull. Korean Math. Soc. 2017 Vol. 54, No. 4, 1141-1158
https://doi.org/10.4134/BKMS.b160180
Published online July 31, 2017
Mohamed Ta\"ib Jakhlouti and Kamel Mazhouda
University of Monastir, University of Monastir
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 $$\sum_{\rho_{a,\chi};\ 0<\gamma_{a,\chi}\leq T}L'\left(\rho_{a,\chi},\chi\right) X^{\rho_{a,\chi}}\ \ \hbox{as}\ \ T\rightarrow \infty,$$ 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$\rm\check{s}$tis \& Steuding \cite{7} and the authors \cite{12}.
Keywords : Dirichlet $L$-function, $a$-points, value-distribution
MSC numbers : 11M06, 11M26, 11M36
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