| It is well-known that number theory is a very important bench of math-ematic which deals with the properties of whole numbers, and the study of sequences is a vital problem in number theory. The research of the properties of mean value of Smarandache sequence was attracted many researchers, and Smarandache sequence is now become more and more popular in the field.In number theory world, there is a famous theorist, professor Florentin Smarandache who's an American-Roumanian. He presents many problems and conjectures about the properties of Smarandache function and sequences which are after his name. In the last century, he published a book named "Only Problems, Not Solutions". In his book, professor Smarandache presented 105 problems and conjectures about the Smarandache function and sequences, there are many researchers studied so hard to find out the answers and solutions about the problems in his book. And they do had many results and theories. The Japanese researcher professor Kenichro Kashihara is one of them, he pub-lished a book named "Comments and Topics On Smarandache Notions And Problems". His study of the mean value of Smarandache sequences is very remarkable.Because of the importance of the Smarandache function and the Smaran-dache sequence in the field of modern number theory and algebra, this disser-tation mainly studied some properties of Smarandache sequences and expanded in some other fields. Use the theories of primary number theory to study the properties of the Smarandache sequences. It is well-known that one of the most important study is the mean value of the Smarandache sequence, it is an im-portant tool in number theory researching. Many famous problems of number theory are closely related to these mean values of sequences. Therefore, any sub-stantiality evolution will be the key power of the Number Theory's evolution.Smarandache sequence is a very good property, the property of expandable. For example, in Banach space. In this dissertation, we will consider the possi-bility that if we can study the properties of Smarandache sequences in Banach space. |
