Research On The Mean Value And Properties Of Smarandache Related Functions

Many scholars have made research on unsolved problems in a book named Onlyproblems,Not solutions!, and they obtained numerous satisfying results. In this paper, wedo further research into the mean value and properties of Smarandache related functionsby using elementary and analytic ways and theories. The specific work has been done asfollows：1.Firstly, this paper is to study the hybrid mean value distribution concerning theSmarandache LCM function SL (n)and the number theoretic function （n）,compositefunction SL (v (n)),as well as the mean value properties of the average variance of SL (n);2.Secondly, we have investigated the hybrid mean value based on the properties ofthe subtraction k-th power part residue function gk(n),the number theoretic function （n）,as well as the divisor function d (n);3.Lastly, we have discussed the mean value properties of the multiplicative functionfr (n)and the arithmetic function u (n),and then we have studied the mean value distributionof the multiplicative function fr (n)in the non-fourth factor sets.