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 $L^{\prime }(s,\chi )$ 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 $$\sum _{{\rho }_{a,\chi };0<{\gamma }_{a,\chi }\le T}{L}^{\prime }\left({\rho }_{a,\chi },\chi \right)X^{{\rho }_{a,\chi }}asT⟶\infty ,$$ where $X$ is a fixed positive number and $\chi$ is a primitive character $\mathrm{mod}q$. This work continues the investigations of Fujii \cite{2,3,4}, Garunk$\check{s}$tis \& Steuding \cite{7} and the authors \cite{12}.

20 pages, to appear at Bulletin of the Korean Mathematical Society

References in corpus (1)

• Numerical study of the derivative of the Riemann zeta function at zeros