Studies On The Mean Value Of Some Arithematical Functions

The number theory is a branch of mathematics which deals with properties of the integers. Many important ideas and methods in modern mathematics are developed constantly as we study the properties of the integers deeply.It is well known that the research of the arithmetical properties of some special sequences and functions is one of the main subjects in the number theory.Some authors had studied them,and obtained some interesting results.The main purpose is mainly use elementary methods to study the properties of a new Smarandache sequence and some special functions,and give some related identities and asymptotic formulas.The main achievements are as follows:1.Smarandache prime additive complement sequence are very important.we mainly study the elementary properties of the Smarandache prime additive complement sequence, and give an important distribution property theorem,an important divergent theorem and an interesting asymptotic formula which involving the sequence.2.Mainly use the elementary methods to study the mean value properties involving the additive k-th power complements b_k(n) and give a sharper asymptotic formula.3.The main purpose is to study two infinity series involving the k-th power complement number and the additive k-th power part residue function f_k(n),and get some interesting identities.