The High Power Mean Of The Exponential Sums And The Diophantine Equation

The exponential sums originally arose in connection with Waring's problem and Goldbach's conjecture, the methods of computing and evaluating the upper bound of exponential sums are called the exponential sums method. The introduction and improvement of the exponential sums method had greatly promoted the development of modern analytic number theory, geometry number theory and the additive number theory (heaps number theory).The research work of evaluating the upper bound and calculating the mean value of the exponential sums is one of the classic content in analytic number theory and has important theoretical significance and application values. Corresponding to the exponential sums is the character sums, it plays a very important role in many famous number theory problems, such as Dirichlet L-function theory, number theory problems relating to arithmetic series, the minimum positive remaining, the minimum primitive root and so on.The calculation and estimation of the exponential sums?the character sums and a lot of their generalization sums are very important not only in the theory research, but also in cryptography. The research of the exponential sums and the character sums have both theoretical significance and practical values. Any substantial progress made in this field will play a major role in the development of number theory and cryptography.Many famous scholars such as L.G. Hua, A.Weil, Gauss, T.Cochrane, etc. had made great contributions to estimate the upper bound of the exponential sums. The single value of the exponential sums is irregular, but the high power means that value owns graceful arithmetical properties. In recent years, many number theory scholars at home and abroad have made further researches into the computation problems of the high power mean of the exponential sums and their generalization sums, Such as the complete triangle sums, the two-term exponential sums, the classic Gauss sums, the quadratic Gauss sums, the character sums and the kloosterman sums etc, and have acquired a lot of research results.Among these existed achievements, various restrictive conditions were made during the process of computing the mean value of the exponential sums, which makes the application limited greatly. This paper will deeply analyze and discuss the calculation problem of the high power mean of the exponential sums and their generalization sums under the more general conditions, extending or improving the research achievements of predecessors. Specifically, the main results are as the following:1. Studied the essential relation between the fourth moment of the two-term exponential sums and the congruence equation, and acquired the precise calculation formula for k?5(mod(p-1)), extending the original research results (k=-1,1,2,3(mod p))2. Studied the fourth power mean of the two-term exponential sums with Dirichlet Character and and acquired the explicit formulae. This extends the results of H.Chen et al. by avoiding the restriction q being an odd integer.3. Studied the sixth power mean of the generalized quadratic Gauss sums when q is an arbitary square-full number, and obtained the precise calculation formulas, generalizing the existing research results.4. Studied the2l-th power mean of generalized k-th quadratic Gauss sums and got the accurate calculation formulas when (n,q)=(k,?q))=1, enriching the existed research results.5. Studied the fourth power mean of the generalized Kloosterman sums for arbitary integers k and q, and obtained the exact calculation formulas, generalizing the existed research results.In addition, we studied the computational problems of the fourth moment of three-term exponential sums, discovered the essential relation between the fourth mean and the system of congruence equations, and several explicit formulas for the fourth power mean of three-term exponential sums are obtained under certain conditions.In this paper, we also introduced several methods about how to find the integer solutions about diophantine equation. By using A.Baker' theory and LLL algorithm, we acquired all the integer solutions of one kind simultaneous Pell equations.