On The Identity Of The Smarandache Function And The Asymptotic Formula

Analytic number theory is a component of number theory,its studying implement is analytic method.Studying the properties of analytic function play an important role in the study of number theory,and they relate to many famous number theoretic problems.Recently,American-Romanian number theorist Florentin Smarandache published a book named "Only problems,Not solutions!" In this book,he presented 105 unsolved arithmetical problems and conjectures about arithmetical functions.Many researchers studied these sequences and functions from this book,and obtained some important valued results on theory.In this dissertation,we use elementary methods and analytic methods to study some problems which were given in "Only problems,Not solutions!",and give some related identities,asymptotic formulae and positive integer solutions of some equations.The main achievements contained in this dissertation are as follows:1.Studying Smarandache function S(n) is very significant.We use the elementary methods to study the solutions of an equation involving the Smarandache functions,getting more general conclusion.An open problem is presented at the same time.2.The Smarandache function has very important positions in the study of number theory.The new Smarandache function G(n) is defined.The elementary properties of G(n) are studied.An exact calculating formula and an inequality involving the new Smarandache function G(n) are given.3.Studying some mean value problems is very significant.We study a hybird mean value problem involving the F.Smarandache LCM function and the divisor function,a mean value of composite function which involving the pseudo-Smarandache functions is given,and obtain some interesting identities and asymptotic formulae for them.