The Equation And Mean Value Of Smarandache Function

Both solutions of Smarandache function equation and the mean value of Smarandache function are important research topics in number theory.Many scholars have carried on the thorough research and constantly put forward some new and valuable problems about number theory and conjecture.This provoked people’s interests for the study of number theory.In this dissertation,we use elementary method and analysis method to study the solvability problem of Smarandache function equations,and the mean value of Smarandache functions.The main content of this dissertation is as follows:In the first part,by studying the number theory function such as Pseudo Smarandache function and the Euler function,we set up the equation Z(nx)= φ(ny),Z(n)= φ2(n)and Z(n)=φe(SL(n)),and all the positive integer solutions of the equation are given.In the second part,we study the mean value of Pseudo Smarandache function Z(n)on simple data set,and the asymptotic formula of(?)Z(n)is given.In the third part,we study the type of(?)δα(n)(SL(n)-P(n))2 presenting the asymptotic formula.In the fourth part,we study the mean square value of Smarandache double factorial function(?)(SDF(n)-SM(n))2,and obtain an asymptotic formula.