Study Of The Nature Of The Smarandache Function And Its Sequence

Number theory is an old mathematical branch, in which it is important to study the mean value and properties of some arithmetical functions and se-quences. American number theorist of Romanian descent Florentin Smarandache proposed 105 unsolved arithmetical problems and conjectures in a book named "Only problems, Not solutions!". After that, many researchers and experts studied these functions and sequences attentively, and obtained some important valued achievements. All of these enrich and improve heavily the development of the number theory.Based on the interests in these above problems, we mainly use elementary and analytic methods to study Smarandache related functions and Smarandache sequences. That is to say, we study the mean value of the Smarandache related function and the convergence and divergence estimates of the Smarandache se-quences. Specifically, the following results are obtained in this dissertation:1. The definition of the function S**(n) relating to the famous Smarandache dual function is given. The mean value of S**(n) is studied by using elementary methods, asymptotic formula of ln([x]!) and the relationship between integration of sinnx and n!!, as well as certain properties of power series, and classifying positive integer n into even and odd. Eventually, a sharper asymptotic formula for it is given.2. The definitions of the Smarandache power function SP(k) and Smaran-dache totien function St(k) are given. The limit about these two functions is discussed by using the elementary and analytic methods. Finally, the result is obtained.3. The definitions of Smarandache alternate consecutive and reverse Fi-bonacci sequence a(n) and Smarandache multiple sequence b(n) are given. The limit and convergence properties for these series are studied by using the ele-mentary method. At the end, the results that and the series is convergent are obtained.