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 the zero-expansion of L'/L', and generalize it to the hybrid mean value case. Some fourth power mean value on B(x) are proposed;Second, we study the mean value of |L'/L'(1,x)|~2k further, and we also study the hybrid mean value of |L'/L'(1,x)| and general Kloosterman sums and general quadratic Gauss sums and a few asymptotic formulae are given;Thirdly, we study Brewer sums and its properties. Using elementary method in number theory, an interesting identity is given;Finally, we work on some special sequences and functions, and give a few sharp asymptotic formulae. The main achievements contained in this dissertation are as follows:1. The study on the problem of the zero-expansion of L'/L' will help us to know more properties of the Dirichlet L-function. In this dissertation, we study the problem of the zero-expansion of L'/L', and obtain an interesting mean value formula about |B(x)|~42. The study on the problem of the hybrid mean value of |L'/L'(1, x)| and we also study the mean value of | L'/L' (1,x)|~2k and general Kloosterman sums and general quadratic Gauss sums and a few asymptotic formulae are given.3. The study of the Brewer sums is one of the hottest issues in analytic number theory. In this dissertation, we use the elementary method in number theory to study Brewer sums and its properties, and an interesting formula is given.4. Some special sequences and functions are studied. We study SCBF function, two new functions and obtain some interesting formulae based on the simple numbers. We also study the distribution properties of the Smarandache m-power complement numbers, Smarandache factorial sequence, and obtain some interesting asymptotic formulae.