On asymptotic properties of the generalized Dirichlet $L$-functions

Abstract

Let $q\ge 3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac{\chi (n)}{(n+a{)}^s},$$ where $s=\sigma +it$ with $\sigma >1$ and $t$ both real. It can be extended to all $s$ by analytic continuation. In this paper, we study the mean value properties of the generalized Dirichlet $L$-functions, and obtain several sharp asymptotic formulae by using analytic method.

15 pages,accepted by IJNT