On The Fourth Power Mean Of Three Term Exponential Sums And Generalized Three Term Exponential Sums

Let p be an odd prime, k, t be positive integers, and let x denote any Dirichlet character mod p. For any integers m,s,n, the three-term exponential sum C?m, s, n, k, t; p? is defined as follows: and the generalized three-term exponential sum C?m, s, n, k, t, x; p? is defined as follows: Where e?y?= e^2?iy. As a special case of the complete trigonometric sum, the three-term exponential sum not only plays a significant role on the study of Waring's problem, but also has a wide range of application in cryptography. Many researchers have studied the various properties of this exponential sums and related sums.The main purpose of this paper is using trigonometric identity and the properties of the reduced residue system to further study the exponential sums. Firstly, we study the fourth power mean of three-term exponential sums p p and give three exact computational formulas when different special values. Then, we study the fourth power mean of gener-alized three-term exponential sums|C?m, s, n, k, t, X;p?|⁴, and obtain a series of well results.