On Of Several F.smarandache Number Theoretic Function

Number theory is a mathematical subject to research the property of integer, as we all know, number theory often is the driving force to promote the development of mathematics in the history, Many important thinking and methods in modem mathematics are developed unceasingly in the study of number theory problem. Moreover arithmetical function's mean value estimated questions is important research subjects, many famous difficult problems closely are related in number theory with it. However, we are very difficult to give the precise formula for mostly arithmetical function, only through the asymptotic formula to open out the change rule and the property.Florentin Smarandache the famous American-Romania mathematician has introduced many very interesting sequences and arithmetical functions, and has proposed a series of outstanding questions and conjecture. In "Only Problems, Not Solutions", Florentin Smarandache made about 105 arithmetical functions and sequences of unresolved questions and conjecture, causing many number theory enthusiasts interested in the study.This thesis has researched some arithmetical function using the knowledge in elementary number theory and analytic number theory. The main results of this paper as following:1. Study new arithmetical function of mean value estimation problem. This section is using the elementary method to study the mean value properties of the arithmetic functionΩ(n-f_k(n)) andδ_m(f_k(n)) on base of the additive k-th Power part residue functionf_k (n), and give some interesting asymptotic formula.2. Research on the properties of F.Smarandache square complementary functionSsc (n). Using elementary method studies the divergent property of the sequence (?)defined with F.Smarandache function Ssc (n), and determines the limit is 1.3. To solve the properties of an equation involving the F.Smarandache function SM (n) and Zw(n). Using of elementary and analytic methods solved the properties of this equation and obtained all positive integer solution to this equation SM (n) = Zw(n).