Some Mean Values On Arithmetical Functions And Equations Involving Arithmetical Functions

It is well known that studying the properties of arithmetical functions is an important part of Number Theory. Various times mathematicians paid great attention to study the properties of integers, and got theoretical results on Number Theory. With the development of the times, some of the old problems have been solved. However, more and more new problems will arise. American-Romanian number theorist Florentin Smarandache introduced hundreds of interesting sequences and arithmetical functions, and presented many problems and conjectures in his life. In 1993, he published a book named "Only problems, Not solutions" . He presented 105 unsolved arithmetical problems and conjectures about these functions and sequences in it. Many researchers studied these sequences and functions from this book, and obtained important results.In this dissertation, we mainly study the mean value problems of some important arithmetical functions, and positive integer solutions of some special equations which involving Smarandache function. The main achievements contained in this dissertation are as follows:1. The main purpose is to study the k-power complement number series, and get two interesting identities. We study the mean value problems of a new arithmetical function, and give two sharper asymptotic formulas.2. we studied the solutions of the equationand got some interesting results.3. Smarandache function S(n) has very important position in the study of number theory. We use the elementary methods to study the solutions of some equations involving the Smarandache function, and give all its positive integer solutions.4. Mainly use the elementary methods to study an equation involving the pseudo-Smarandache squarefree function, and get some very interesting results.