Several Special Number Theoretic Function

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 progress in this field will contributeto the development of analytic number theory. American-Romanian numbertheorist 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 problems and conjectures about arithmetical functions and sequences in it. Many researchers studied these sequences and functions from this book, and obtained lots of important results.In this dissertation, we study the mean value problem of a special arithmetical function and some aspect about the Smarandache unsolved problems. The main achievements contained in this dissertation are as follows:1. The mean value problem of a special arithmetical function are studied. We study the properties of a compound function which contain e_p(n) and S_k(n) and give an asymptotic formula while e_p(n) denotes the largest nonnegative integerαsatisfying p~α| n and (?) denotes the largest positive integer m satisfying m~k| n.2. The Smarandache Ceil function S_k(n) has a very important position in the study of number theory. We use the elementary methods to study the solutions of an equation involving the Smarandache Ceil function.3. Studying some infinity series is very significant. We use the elementary method to study the convergent properties of some special functions, and give some interesting identities.