It is well known that the study of arithmetical functions on some special sequences play an important role in the study of analytic number theory,and they relate to many famous number theoretic problems.Any nontrivial progress in this field will contribute to the development of analytic number theory.Professor F.Smarandache is a Romanian famous number theorist expert. One of his numerous contributions is the hundreds of interesting sequences and arithmetical functions that are introduced,and presented many excellent unsolved problems and conjectures in his life.He published a book named "Only Problems ,Not Solutions".He presented 105 unsolved arithmetical problems and conjectures about these functions and sequences in it,it arose great interests for scholars.Many researchers studied these sequences and functions from this book, and obtained important results.In this dissertation,we use the elementary methods and analytic methods to study the mean value problems of some important arithmetical functions and introduce a new arithmetical function.The main achievements contained in this dissertation are as follows:1.The mean value problems of some Smarandache functions are studied,and an asymptotic formula for its mean value is given.2.Some properties of the integer part of the m-th root and the largest mth power not exceeding n using the elementary methods are discussed,and some interesting identities involving these numbers are given.3.Another Smarandache dual function S^{**}（n）is introduced.The convergence of an infinity series involving S^{**}（n）is studied by the elementary methods and an interesting identity for it is proved. |