The Related Issues Research Of Pseudo Smarandache Function And Euler Function

Pseudo Smarandache function and Euler function have a significant position in the field of number theory.Many number theory scholars have carried on the thorough research,and put forward some valuable research problems,stimulated the interests of people to explore continuously.Based on the interests of pseudo Smarandache function and Euler function,we use elementary and analytical methods to study the mean value problems of pseudo Smarandache,the equation solvability and the mean value distribution problems of Euler function by reading a large number of Chinese and foreign literatures.Including the following aspects:In the first part,we discuss the mean value problems of the composite function about the pseudo Smarandache function and the arithmetical function(?)(n),the number theory function D(n),the Pseudo-Smarandache-Squarefree function Zw(n),Smarandache LCM functionSL(n),and give sharper asymptotic formulas of mean value distribution.In the second part,firstly we discuss solvability problem of the equation φ(xyz)=5(φ(x)+ φ(y)+φ(z))including Euler function,and give all its solutions.Secondly,we study the equation solvability problem of Euler function and the Pseudo-Smarandache-Squarefree function.Finally,we study the mean value problem of the Euler function and the Pseudo-Smarandache-Squarefree function,and give sharper asymptotic formula of the mean value distribution.