On The Mean Value Calculation Problems Of Arithmetical Functions

It is well known that the mean value problems of arithmetical functions play an important role in the study of the 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 F. 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 of this book, and obtained many important results.In this dissertation, we study the mean value problems of some important arithmetical functions, and study the solutions of some equations involving the Smarandache dual function. We obtain some sharper asymptotic formulas and some exact calculating formulas for them. The main achievements contained in this dissertation are as follows:1. In the third Chapter, we use the elementary methods to study the solutions of some equations involving the Smarandache dual function, and give the positive integer solutions of them.2. In the forth Chapter, we study the mean value of properties of Smarandache factorial function and it' s mean value, using analytic methods give several interesting asymptotic formula for the reciprocal of S_k(t,n) function.3. For any positive integer n, about the Smarandache LCM dual function SL*(n), the main purpose of the fifth Chapter is using the elementary method to study the calculating problem of a Dirichlet series involving the Smarandache LCM dual function SL*(n) and the mean value distribution property of SL*(n), obtain an exact calculating formula and a sharper asymptotic formula for it.