About Some Mean Value Properties Of The Smarandache Function

It is well know that the problems of mean value of the number theory function play an important role in the study of number theory, and many well-known problems of number theory are closely related. Therefore, any nontrivial progress in this field will contribute to the development of number theory. In this dissertation, we study the mean value problems and nature of some number theory functions. About the problem of mean value, Firstly, we studythe Smarandache function Sdf(n), and get a sharper asymptotic formula. Secondly, we studythe mean values of digital sum function raised by Samarandache Professor, and a precise formula is given. Thirdly, we study the mean value of counting function of the Fibonacci series and Lucas series, making some specific formulae. Defining a broad series of Lucas and discussing some properties about it. The main consequents involved in this paper are as follows:1. The research on the mean value of Smarandache function play a very important role in number theory, a lot of number theoretic functions is closely related to the mean value. Themean value of Smarandache function Sdf(n) is studied in Chapter II of this paper with theelementary and analytic methods, and a sharper asymptotic formula is given.2. The study on Smarandache function is very rich, the mean value of digital sum function raised by Samarandache Professor is studied in Chapter III of this paper, and a precise formula is given.3. It is very interesting to do some research on mean value of special series, Well-known Fibonacci series and the Lucas series play an important role in the theory of mathematical research, the mean value of counting function of two types are studied with elementary methods in Chapter IV of this paper, drawing some new mean value formulas, and discussing some properties of a broad series.