About The Mean Estimate Of The Number Theoretic Function And Smarandache Problems

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. American-Romanian number theorist Florentin Smarandache introduced hundreds of interesting sequences and arithmetical functions, and presented many problems and conjectures in his life. In 1991, 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 study the mean value problems of some important arithmetical functions and some aspect about Smarandache unsolved problems. The main achievements contained in this dissertation are as follows:1. The mean value problems of some functions are studied. We study the properties of this sequence e_p(n) and give some asymptotic formulas while e_p(n) denoted the largest exponent of power p which divides n. We introduce two new arithmetical functions P_d(n) and f(n), and give two interesting asymptotic formula.2. 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.3. We study the mean value properties of the second Smarandache pseudo-odd number sequence and pseudo-even number sequence, and give some interesting asymptotic formulae for them.4. Studying some infinity series is very significant. We use the elementary method to study the convergent properties of this simple infinity series, and give some interesting identity.