The Research On The Related Problems About Euler Function Equation And Smarandache LCM Function

The function equation and function mean value have been the core in number theory research.In recent years,Euler function equation and the mean property of Smarandache functions have been studied by quite a few scholars,and made a lot of important theoretical value achievements,it played an important role in the development of number theory.Based on the interest of the above problems,several solutions of Euler function equations are studied by means of elementary methods and analytic methods,and mean value of the function of Smarandache LCM is given,and the following results are obtained:1 All positive integer solutions of Euler function equation φ(xy)= 7φ(x)+ 9φ(y) are given.2 All positive integer solutions of Euler function equation φ(abcd)=k(φ(a)+φ(b)+ φ(c)+ φ(d))are given,when the k-1,2.3 All positive integer solutions of the arithmetic function equation S(SL(n2))=φε(n)are given,when the ε =1,2.4 A aybrid mean value problem of the function(SL(n)-SM(n))β is studied and an interesting asymptotic formula is given by using the elementary method and the distribution property of prime numbers.