On Dirichlet Characters of Polynomials

summary:Let $q \ge 3$ be a positive integer. For any integers $m$ and $n$, the two-term exponential sum $C(m,n,k;q)$ is defined by $C(m,n,k;q) = \sum _{a=1}^q e ({(ma^k +na)}/{q})$, where $e(y)={\rm e}^{2\pi {\rm i} y}$. In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it