On a Kind of Dirichlet Character Sums

Abstract

Let $p\ge \mathrm{3}$ be a prime and let $\chi$ denote the Dirichlet character modulo $p$. For any prime $q$ with , define the set $E\left(q,p\right)=\left\{a\mid 1\le a,\stackrel{-}{a}\le p,a\stackrel{-}{a}\equiv 1\left(\text{mod}p\right)\text{and}a\equiv \stackrel{-}{a}\left(\text{mod}q\right)\right\}$. In this paper, we study a kind of mean value of Dirichlet character sums $\sum a\le pa\in E\left(q,p\right)\chi (a)$, and use the properties of the Dirichlet $L$-functions and generalized Kloosterman sums to obtain an interesting estimate.