Mean Of Arithmetic Functions

Mean value problems of arithmetical functions play an important role in the study of analytic number theory, and they related to many famous number theoretic problems. Any nontrivial progress in this field will contribute to the development of analytic number theory.Professor F. Smarandache is a Rumanian famous number-theoretic expert. One of his numerous contributions is the excellent unsolved problems that are presented by him continually. In 1993, Professor F. Smarandache presented 105 unsolved problems in <>, it arose great interests for scholars.In this dissertation, we use the elementary methods and analytic methods to study some problems which were given in <>, especially to study the mean value problems of some arithmetical functions, and give several interesting asymptotic formulae. The main achievements contained in this dissertation are as follows:1. In the first problem, we study the mean value of primitive numbers of power P and giving a the asymptotic formulae for it. we also study the properties of primitive numbers of power P, and give a triangle inequality for it.2. In the second problem, study the mean value properties of the additive k-th power complements, and give some interesting asymptotic formulae for it.3. The study the mean value of Smarandache ceil function and dual Smarandache ceil function's hybrid mean value with cube complements, and we obtained some exact asymptotic formulae for it.4. We use the elementary methods to study the new function S_k(n) and G_k{n) about n, and several interesting asymptotic formulae are obtained.