The Mixed Mean Value Of Generalized Dedekind And Binary Exponential Sum And The Problem Of A Class Of Diophantine Equation

Number theory is the subject of studing the various properties of integet, which is called the Mathematics crown is beautiful branch of Mathematics fields. As a highly abstract subject, reasearching and solving the many problems of Number theory has the far-reaching significance. The value mean distribution problems about some famous sum forms in number theory is always the contre content of number theory study.This paper mainly studies the hybrid value including the generalized Dedekind sums and the two-term exponential sums and a class of number the-oretic function equation by using the elementary and analytic methods, The specific research results are as follows:1. Using the relationship between the generalized Dedekind sums, the two-term exponential sums, Dirichlet L-function and charater sums and the es-timation of Dirichlet series, study the hybrid value including the generalized Dedekind sums and the two-term exponential sums, and give some sharper asymptotic formulas.2. Through the study of exponential Diophantine equation xy+yx= zz, some properties of the least solution of Pell’s equation, we prove that the equation xy+yx= zz has no positive integer solutions (x, y, z) with x and y being odd primes.