On the mean value of the generalized Dirichlet L-functions with the weight of the Gauss Sums

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. For any integer $m$, the famous Gauss sum $G(m,\chi )$ is defined as follows: $$G(m,\chi )=\sum _{a=1}^q\chi (a)e\left(\frac{am}{q}\right),$$ where $e(y)=e^{2\pi iy}$. The main purpose of this paper is to use the analytic method to study the mean value properties of the generalized Dirichlet $L$-functions with the weight of the Gauss Sums, and obtain a sharp asymptotic formula.

14 pages