On The Mean Value Of Some Arithmetical Functions

The mean value problems of functions play an important role in the study of analytic number theory, and they are related to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory.American-Romanian number theorist Florentin Smarandache introduced hundreds of interesting sequences and arithmetical function, and presented many problems and conjectures in his life. In 1993, he published a book named "Only Problems, Not Solutions!". The book presented 105 unsolved arithmetical problems and conjectures about these functions and sequences. Many researchers have studied these sequences and functions from this book, and obtained some important results.In this dissertation, we studied the mean value and hybrid mean value properties of some arithmetic function, and got some asymptotic formulas about them. Specially, the main contributions in this paper are as follows:1. Using the analytic methods, we studied a limit problem involving the F.Smarandache square complementary number Ssc(n), and obtained its limit value.2. We studied the asymptotic properties of the integer part of the k-th root positive integer and gave an interesting asymptotic formula.3. Using the elementary methods, we studied the hybrid mean value involving Smarandache function and the least prime divisor function, and gave an asymptotic formula.