On The Mean Value Of Some Arithmetical Functions

It is well known that the Euler function, Smarandache function and the Riemann-zeta function play an very important role in the study of number theory. Therefore, It has great significance to reveal the inherent relation among the Euler function, the Smarandache function and the Riemann-zeta function.In this dissertation, we study the mean value problem of some important arithmetical function. The main achievements contained in this dissertation are as follows:1. Euler function φ(n) and Smarandache multiplicative function S₁(n) have very important position in the study of number theory. We study the relations between the Euler function φ(n) and the Smarandache multiplicative function S₁(n) , found the equation φ(n) = S₁(n), and obtain its all positive integer solutions.2. Riemann-zeta function is essential in the study of number theory. In this dissertation, using the property of Smarandache multiplicative function S₁(n), we discuss the relation between Riemann-zeta function and Smarandache multiplicative function S₁(n), and obtain some important identity about them.3. In this paper, using elementary method, we study the mean value properties of the triangular numbers part residue sequence and obtain an interesting asymptotic formula for function a(n) and d(a(n)).