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.