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

Rong Ma, Yana Niu

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,χ,a)=\sum_{n=1}^{\inf…