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 . The main achievements contained in this dissertation are as follows:1. let a_k(n) denotes the smallest positive integer such that a_k(n) + n is a polygon number. In the third chapter, we mainly study the properties of smarandache polygon number by combining elementary methods with analytic methods . As a result, we got a series of interesting mean value formulae of Smarandache functions.2. The great common Divisors theory play an important role in the number theory . In the fifth chapter, we use the properties of the great common Divisors and the least common multiple theory to get some formula of the ratio of two least common multiple .3. Let S(n) denote the smallest number m such that n| m!. In this chapter, We study the mean value of smarandache factorial function. and an interesting asymptotic formulas are given.