This article applied the properties of character sums, quadratic character, and classical Gauss sums to study the calculations of the hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums. It also provided exact formulas for calculating these hybrid power means.

In this paper, based on the analytic method and the properties of Gauss sums, we study the computational problems of the fourth power mean value of one kind two-term exponential sums through the classification and estimation of Dirichlet characters and give it a calculation formula or asymptotic formula in different conditions.

The calculation of the mean value of exponential sums C(m, n, r, s; q) and its upper bound estimation is closely related to many number theory problems, such as Waring’s problem. Let p be an odd prime, this paper focuses on the problem of the calculation of the fourth power mean value of one kind two-term exponential sums when parameter n=1 and exponent r=4, s=2. By utilizing the analytic methods, the odd-even property and orthogonality of Dirichlet character, and the properties of the character sums, we study the calculating problem of the fourth power mean of the two-term exponential sums for the form as C(m, 1, 4, 2; p), and give an exact calculating formula for it when the prime p≡3 mod 4. At the same time, for such research content, some open problems in this field are also proposed.