Mean Value On Some Arithmetical Functions And Solvability Of The Functional Equations

It is well known that the mean value problems of arithmetical functions play 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.Professor F.Smarandache is a Romanian famous number theorist. One of his numerous contributions is the excellent unsolved problems that are presented by him continually. 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 in this book, and obtained many important results.In this dissertation, we use the elementary methods and analytic methods to study some problems which were given in "Only Problems, Not Solutions!" and "Unsolved Problems in Number Theory" , especially to study the arithmetic properties on some special sequences, and give several interesting asymptotic formulae. The main achievements contained in this dissertation are as follows:1. In the first problem, we use the elementary method to study the mean value problems of the F.Smarandache square complementary number, and obtain its limit value.2. In the second problem, we use the elementary methods to study the asymptotic properties of S_k(n), and give an interesting asymptotic formula for it.3. We study a kind of function by using the elementary method, and get all real solutions for it.