Estimated Mean Of Arithmetic Functions

It is well known that the mean value problems of arithmetical functions play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory.In this dissertation, we study the mean value problems of some important arithmetical functions. Firstly, we study the problem of an integer and its inverse, and generalize it to the multiple varieties and high dimensions case. Some high dimensions mean value and hybrid mean value are proposed; Secondly, we study the hybrid mean value between the hyper Cochrane sums and hyper Kloosterman sums; Thirdly, we generalize several important identities involving the character sums of polynomials; Fourthly, we study the mean value of the two-term exponential sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean; Fifthly, we work on some special sequences, and give a few sharp asymptotic formulae. The main achievements contained in this dissertation are as follows:1. The study on the problem of an integer and its inverse will help us to know more properties of the distributions of integers. In this dissertation, we study the problem of an integer and its inverse, generalizing it to the multiple varieties and high dimensions case, and obtain some hybrid mean value formulae.2. The Dedekind sums and Kloosterman sums enjoy their long history. We study the Cochrane sums, which is analogous to the Dedekind sums, and generalize it to the multiple varieties case. The hybrid mean value between the hyper Cochrane sums and hyper Kloosterman sums is studied, and an interesting asymptotic formula is given.3. The study of the character sums is one of the hottest issues in analytic number theory. In this dissertation, we generalize several important identities involving the character sums of polynomials.4. The exponential sums also enjoys its long history. In this dissertation, we study the mean value of the two-term exponential sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean.5. Some special sequences are studied. We study the distribution properties of the m-power residues and k-power free numbers, and obtain some interesting asymptotic formulae; We also study a new counting function involving the famous Fibonacci numbers, and an exact calculating formula for it is given.