The Arithmetic Functions Mean And Diophantine Equation Solvability Study

Arithmetic function is defined as real value function or complex value func-tion in positive integer set. It is one of the most widely used concept in number theory. Arithmetic function such as Dedekind sums, Gauss sums, Kloosterman sums, Cochrane sums, Dirichlet L-function and D.H.Lehmer problems plays a very important role, and is closely related to many famous problems in analytic number theory, so the study to them is very meaningful. Moreover, the methods, skills and conclusions to study these functions also can be used in cryptography information such as field.In recent years, the research results about the mean value properties, the error estimate, the upper bound estimate, the exponential sums, the character sums and the extended form of these arithmetic functions are more abundant, which make significant progress in this research field and plays an important role in the development of analytic theory, what is more of very realistic significance.The exponential Lebesgue-Nagell equation is also a recent important re-search in number theory, especially solving it when p takes different values has been a hotspot.Calculation of polynomial functions plays an important role in theory and application of mathematical. Although there are many scholars studied and gained a lot of meaningful results, however, the integral calculation of the poly-nomial functions is rare, particularly studing the integral calculation combined with Fibonacci numbers and Lucas numbers is more rare.Based on the above development present situation, in this paper, we study the asymptotic properties of classical sums and mean value in analytic theory, the exponential Lebesgue-Nagell equation and its integer solutions, and the integral calculating problem involving the polynomial function in combinatorial number theory. Specially, the main achievements contained in this dissertation are as follows:1. Reseach on the Dedekind sums and the Lehmer's problem.Using the properties of Dedekind sums and the mean value theorem of Dirichlet L-functions to study the hybrid mean value problem involving the error term of Lehmer's problem and the Dedekind sums, and establish a sharp asymptotic formula for it.2. Reseach on the mean value involving Dedekind sums and two-term ex-ponential sums.Using the analytic methods to study the mean value properties involving the classical Dedekind sums and two-term exponential sums, and give three sharper asymptotic formulae for it.3. Reseach on the exponential equation and its integer solutions.Using the elementary method to study a complete classification of all pos-itive integer solutions (x, y, m, n) of the exponential Lebesgue-Nagell equation x2+p2m=yn, gcd(x,y)=1, n>2.4. Reseach on the integral calculation of polynomial function.Using the elementary method and the properties of the Legendre polynomi-als to study an integral calculating problem involving the Legendre polynomials. At the same time, give an interesting calculating formula.