Mean Value On Some F.Smarandache Simple Functions

Analytic number theory is a component of number theory,its studying implement is analytic method.Mean value problems of some arithmetical functions play an important role in the study of analytic number theory,and they related to many famous arithmetical problems.Therefore,to study their properties is very significate.Professor F.Smarandache is a Romanian famous number theoretic expert.In 1993,Professor F.Smarandache presented 105 unsolved problems in "Only Problems, Not Solutions",most of them are related to number theory.In this dissertation,on the basis of my interesting in the F.Smarandache function,we used the elementary methods and analytic methods to study the properties of some special functions,and set up some new functions by using the familiar functions.In details,the main achievements contained in this dissertation are as follows:1.Studying the properties of the fnnction p(n),and getting the asymptotic formula of d(p(n))by using the elementary method.2.By studying the properties of Smarandache simple function,we studying the mean value ofσ_α(p(n))which is the generalization of the function of d(p(n)).3.By using the analytic method we study the function SL(n)andσ_α(n), then obtained some interesting asymptotic formulae.4.By studying the properties of Smarandache simple function and F.Smarandache LCM function,using the prime number theorem and Abel identity to study the mean value of SL(Z(n)),and obtained some interesting asymptotic formulae.