Estimates For The Mean Value Of Two Types Of Arithmetic Functions

It is well known that mean value properties of arithmetic functions play an important role in the study of analytic number theory. Exponential sums are useful tools to deal with the problems arisen from arithmetic functions, and they relate to many famous number theoretic problems. Kloosterman sum, as a special exponential sum, has been studied widely not only on Diophantine equation and modular form in number theory, but also on the classification problems of Bent functions in cryptography. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory.In this dissertation, the mean value of divisor function defined on powers or powers of the divisor function and the estimation of an upper bound for incomplete high-dimensional Kloosterman sums are studied via applying the elementary and analytic methods.Our dissertation is divided into three parts.In the first chapter, we introduce the background and significance of research topics of this paper. Besides we explain the main ideas and lemmas in this dissertation, and we give the main conclusions.In the second chapter, we utilize properties of exponential sum and character to obtain the estimation of an upper bound for incomplete high-dimensional Kloosterman sums.In the third chapter, we study the mean value of divisor function defined on powers or powers of the divisor function by use of the properties of Riemann Zeta function.