Research On The Properties Of Smarandache Functions

The Smarandache function is one of the important contents of the number theory.With the continuous exploration and research of it,there appears many types of functional equations and research methods.This thesis is based on a large number of books and literatures related to Smarandache problems,mainly using elementary and analytical methods to study several functions of Smarandache issues,the specific content is as follows:Firstly,the problem of solvability of two equations S(SL(n))?(?)(n)and Z(SL(n))-(?)(n)involving the function of Smarandache LCM when e = 1,2 is discussed and all the positive integer solutions of the equation are given.Secondly,the mean value of Smarandache LCM function on the simple sequenceis studied,and the asymptotic formulas of(?)and(?)are obtained.Atthe same time,the ?-th mean distribution of Smarandache LCM function SL(n)and the maximum prime factor function is discussed.Next,the?-th hybrid mean between Smarandache multiplicative function and the maximum prime factor function is studied.Finally,an interesting asymptotic formula of(?)and(?)respectively is given.At last,we study the lower bound estimate problem of the Smarandache LCM function SL(n)on special sequence 2P ± 1.It is given and proved SL(2p±1)?10p+1,when p?17 be any prime.