On The Mean Values Of The Euler Function Of Positive Integers With Square-free

Studying the mean values of the Euler function of a number function is a classic question in number theory. Let cp(n)denote the Euler function defined to be the number of positive integers n which are coprime to n.In many number theoretic problems,the following sum often has to be considered: (?)For example,Goldbach’s problem, Waring’s problem. An asymptotic formula for this sum will be given. The method is to use the given result of (?) and (?)The process calculates by the multiplicative function and categorizes the situations according to the parameter value of a and b.