Number Theory Function Mean Estimate

It is well known that the mean value problems of number theory 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 number theory.In this dissertation, we study the mean value problems of some special number theory functions, relation with some important functions and a series of asymptotic formulas about some hybrid functions. The main achievements contained in this dissertation are as follows:1. The study on the mean value of Smarandache functions play an important role in the study of number theory, and they relate to many famous number theoretic problems. In this dissertation, we study the mean value of some special number theory functions, an identity and mean value between Smarandache function and mobius function, Smarandache symmetric sequence and it' s properties, using elementary and analytic methods got a series of new asymptotic formulas.2. Additive analogue of smarandache functions enjoy their long history. In this chapter, We study the mean value of additive analogue of smarandache single factorial function and smarandache double factorial function, and several interesting asymptotic formulas are given.3. Properties of the reciprocal of δ_k(n) function and it' s mean value are study, using analytic methods give an exact calculating formula for the reciprocal of δ_k(n) function.