Some Asymptotic Formulae Involving The Smarandache Functions And Sequences

It is well known that the arithmetical properties of some special sequences and functions play an important role in the study of number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of elementary number theory and analytic number theory.American-Romanian number theorist Florentin Smarandache presented many problems and conjectures on special sequences and arithmetical functions. In 1993, he published a book named " Only problems, Not solutions!" in Xiquan Publishing House in American. In this book, he presented 105 unsolved arithmetical problems and conjectures about special sequences and arithmetical functions. Many researchers studied these sequences and functions from this book, and obtained some important valued results on theory. So dose another famous book " Unsolved Problem in Number Theory" , written by R.K.Guy from Canada.In this dissertation, we use elementary methods and analytic methods to study some problems which were given in "Only problems, Not solutions!" and "Unsolved Problem in Number Theory" , especially to study the arithmetical properties of some new Smarandache sequences and functions, 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. Properties of the Smarandache Least Common Multiply(LCM) sequence and some related sequences are studied, some interesting identities, deeply asymptotic formulae and new limit theorems which involving these sequences are given.2. Smarandache function S(n) and Smarandache power function SP(n) have very important positions in the study of number theory. We use the elementary methods to study the solutions of some equations involving the Smarandache functions, at the same time, we presented some open problems.3. Studying some infinity series is very significant. We use the elementary method to study the properties of infinity series involving Smarandache function e_p(n) and the irrational root sieve sequence, and give some interesting identities and asymptotic formulae for them.