On Weighted Mean Of Exponential Sum And Its Application

The study on the weighted mean value of exponential sums and its appli-cations is one of the important issues of the research in number theory. The ex-ponential sums、character sums、Dedekind sums、Cochrane sums、Gauss sums‘Kloosterman sums etc. of analytic number theory has a long history and rich content, and there’s also a close link between them. In recent years, many scholars had studied those problems deeply, and obtained many inter-esting results. Undoubtedly, these research played a significant role in the development of the field of number theory.Based on the interests in these questions, the main purpose of this disser-tation is to study the computational problems of higher mean value of the two exponential sums and the hybrid mean value of Dedekind sums, Cochrane sums and other triangle sums, and obtain some identities and asymptotic formula. Also, using elementary methods and the floor function to the re-ciprocals of the sums, we obtain a series new and interesting identities about the famous numbers. The main achievements contained dissertation are as follows:1、Researches on the high power mean of the mixed exponential sums: using the elementary methods, algebraic methods and the theory of complex functions, we study the computational problems of the fourth and sixth power mean of the mixed exponential sums and the fourth power mean of the gener-alized two-term exponential sums, and give some exact computation formula and conversion formulae for them;2、Researches on the hybrid mean value related to Dedekind sums and other triangle sums:we use the definitions、the properties of the Dedekind sums、two-term exponential sums、quadratic Gauss sums and analytic meth- ods to build the relationship between the classical Dedekind sums and two-term exponential sums、the classical Dedekind sums and the quadratic Gauss sums respectively, study their mean value computational problems, and obtain some interesting computational formulae and asymptotic formula for them;3, Researches on the hybrid mean value related to Cochrane sums and other triangle sums:using the elementary methods, the analytic meth-ods, the definitions and the properties of the Cochrane sums, two-term exponential sums、Kloosterman sums, we study the mean value computa-tional problems of the Cochrane sums weighted by two-term exponential sums、Cochrane sums and Kloosterman sums respectively, and obtain some exact computational formulae and asymptotic formula for them;4、Researches on the infinite sums derived from the reciprocals of the famous numbers in number theory:applying the floor function to the recip-rocals of the sums, we obtain a series new and interesting identities involving the the Fibonacci numbers、Lucas numbers and Pell numbers;5、Researches on some sums of powers of Fibonacci polynomials and Lucas polynomials and the computational problem of Dedekind sums and second-order linear recurrence sequence, and gives several exact calculating formulaes for them. Also, using these identities we shall prove a conjecture proposed by R. S.Melham.