The Mean Value Problems Of Some Special Number Theory Function And All Solutions Of The Equation

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 studied the mean value and the hybrid mean value properties of some arithmetic functions, and got a series of asymptotic formulae about them; we set up some equations by studying the relation of some functions, and solved them completely; we obtained the identities involving Lucas numbers, the main achievements contained in this dissertation are as follows:1.For convenience, this article gives the chapter needed to research the background and significance of the topic, describes the study of the status quo at home and abroad, and then gives some of the major results of research for this paper has done a study on the bedding. 2.we study the mean value problems of some special number theory functions by combining elementary methods with analytic methods. As a result, we got a series of interesting mean value formulate of Smarandache functions.3.By studying the properties of Smarandache function, Euler function, we set up the equation (?)SM(d) = (?)SL(d), SM(1~2)+SM(2~2)+…+SM(n~2) = SM((?))and (?)(d) =φ(n),and solve them completely,got all positive inter solutions forthem.