| The solution of the function equation of number theory and it’s mean value is a classical and important research topic in number theory,which is favored by the Scholars of number theory,and has also obtained a series of good results.These results lay the groundwork for the further study of number theoretic functions.In this thesis,we mainly use the typical elementary methods and analytic methods to study the solution of the function equation of number theory and the mean value problem related to Smarandache LCM function.The main results are as follows:The first part:with elementary method,the problem of positive integer Solutions of Euler function equation is discussed.Such as nonlinear Euler function equation φ(ab)=5φ(a)+8φ(b)+6 and ternary variable coefficient hybrid Euler function equationφ(abc)=mφ(a)φ(b)+nφ(c)when(m,n)=(2,8),(3,4),then all positive integer solutions of them are given.The second part:with elementary method,the number theory function equation Z(n)=cp2(SL(n))and Z(n2)=φe(SL(n2))when e=1,2 are analyzed.And then all positive integer solutions of them are obtained.The third part:with elementary and analytic method,the mixed mean properties of Smarandache LCM function SL(n)and it’s dual function SL*(n),Pseudo Smarandache function Z(n)and function W(n)are studied,and three interesting progressive formulos for compand function SL2(n)·Z(n),SL(W(n))·SL*(W(n))and(?)are given. |
