The Nature Of The Equation Involving The Smarandache Function And Mean Problems

As we all know, American-Romanian number theorist Florentin Smaran-dache presented a lot of unsolved problems and conjectures on Smarandache functions. In the book "Only Problems, Not Solutions! ",105 unsolved arith-metical problems and conjectures were presented. Although many researchers answered these unsolved problems mostly, and obtained some good valued re-sults about problems on theories, many unsolved new problems still exist. At the same time, many problems have good values on theories and all of this inspire the reader's interesting extremely.In this dissertation, elementary and analytical methods were applied to de-liberate some new Smarandache functions and the arithmetical sequences which were given in the book "Sequences of Numbers Involved in Unsolved Prob-lems". At the same time, some related identities and asymptotic formulas were given. The main positive results contained in the dissertation are:1. It is very significant to deliberate some infinity series. In this section, the properties about infinity series involved in the Smarandache quotients sequence {Q(n)} are studied, and we also gave an identity involved in the Smarandache quotients sequence.2. Mean value is very important in number theory. We used elementary methods to discuss the mean value about the Smarandache functions epm(n) and Rieman Zeta function, and we also obtained some interesting identities and asymptotic formula.3. In this part, an interesting mean value formula about the Smarandache double factorial function SDF(n) were studied. We used analytic methods to obtain an interesting mean value.