Mean Value And Convergence On Some Arithmetical Functions

A real-valued or complex-valued function defined on the positive integers is called an arithmetical function or a number-theoretic function. The research on mean value of arithmetical functions is an important subject in the study of the Number Theory, and the solving of many other mathematical prob-lems can be attributed to counting function of the solution. With the in-depth study of the integral development, many important ideas, methods and concept of modern mathematics have been continually developed.In the 19th century, with the formation of Analytic Number Theory, the research on mean value of arithmetical functions plays an increasingly important role in the development of Analytic Number Theory.In this dissertation, we study the mean value problems of some important arithmetical functions and some aspects about Smarandache related problems. The main achievements contained in this dissertation are as follows:1. The convergence problems of some functions are studied, and the propertiesof functions sum from n=1 to∞((-1)~n)/((Z(n))~s) and sum from n=1 to∞((-1)~n)/((Z_*(n))~s) are discussed, and some asymptotic formulas are given.2. The properties of the sequence {e_p(n)} and some interesting asymptotic formulas for (?) e_p~m(n) are obtained where p is a prime, and e_p(n) denote thelargest exponent of power p which divides n.