On The Smarandache Function, The Lower Bound Is Estimated On The Specific Sequence And Its Associated Problems

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 num-ber theory and analytic number theory. American-Romanian number theorist Florentin Smarandache presented many problems and conjectures on special sequences and arithmetical functions. In his book-"Only problems, Not solu-tions", published in Xiquan Publishing House in American, he put forward 105 problems and conjectures on number-theoretic function and sequence. Mean-while, in the book of "Comments and Topics On Smarandache Notions and Problems", Doctor Kenichiro Kashihara from Japan also brought forth a lot of problems on Smarandache function, interested a wide mathematical enthusiasts on number theory. Many researchers studied these sequences and functions from this book, and obtained some important valued results on theory. In this disser-tation, we use elementary methods and analytic methods to study some related problems and get some better results. The main achievements contained in this dissertation including the following three aspects:1. We used elementary method to study the low bound estimate of the Smarandache function at some special sequences. Obtained some lower bound estimate.2. Used the elementary method, analytic method and the prime distribution theorem to study the distrbution properties of a mean square value involving the Smarandache LCM function and its dual function. Given a sharpe asymptotic formula for the mean square value of the Smarandache LCM function and its dual function. Obtained some interesting mean square value distribution properties of the Smarandache LCM function and its dual function.3. We defined a new Smarandache power k sieve, use the elementary method to study the properties of the k-ary power sieve sequences. Obtained an inter-esting asymptotic formula.