New Mixed Exponential Sums and Their Application

Abstract

The main purpose of this paper is to introduce a new mixed exponential sums and then use the analytic methods and the properties of Gauss sums to study the computational problems of the mean value involving these sums and give an interesting computational formula and a sharp upper bound estimate for these mixed exponential sums. As an application, we give a new asymptotic formula for the fourth power mean of Dirichlet $L$-functions with the weight of these mixed exponential sums.