About Some Mean Value Properties Of Arithmetic Functions

The mean value problems of arithmetical functions plays an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory. But for most arithmetic function, it is difficult to give an exact calculate formula. So many people shew deep interesting in how to give a precise formula. In past years, a large number of works have been done in these and achieved a good achievement. The main achievements of this paper are the mean value properties of some arithmetic functions and the existence of solution on the equation about the square complement number, and obtained some interesting conclusions. Specially which contained in this dissertation are as follows:1. Some asymptotic formulae on the Smarandache Pseudo-number sequences;2. Studying the numbers of the solution of the equation for the square complements, and giving the all solutions of the equation;3. We studied the mean value properties of a arithmetic function involving the integer part of the M—th root of an integer and theκ—th power free numbers, and some interesting asymptotic formulae were given.