A Guess On The Smarandache Multiplicative Function

It is well known that the Smarandache problems play an important role in the study of number theory, and they relate to many famous number theoretic problems. Therefore, any fundamental progress in this field will contribute to the development of elementary number theory. American-Romanian famous number theorist Florentin Smarandache introduced hundreds of interesting sequences and number theory functions, and presented many problems and conjectures in his life. In 1991, he published a book entitled "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 many important results.This dissertation mainly studied some aspects about Smarandache problems. The main achievements contained in this dissertation are as follows:1. One conjecture about Smarandache function S (n) is studied and partly solved, and some results that help to illustrate the correctness of the conjecture are generated.2. On the basis of the Smarandache function S(n), a new Smarandache multiplicative function (?)(n) is introduced, that is, (?)(1) =1, for any integern, if n = p₁^α₁p₂^α₂…p_s^α_s is the standard decomposition of n, then (?)(n) = max {(?)(p₁^α₁), (?)(p₂^α₂),…,(?)(p_s^α_s)} is correct. For the prime power order, a function(?)(p^α) =αp is introduced. Obviously, this is a new Smarandache multiplicativefunction, the sum function of the factors of its reciprocal can be written as Smd(n) = (?)1/(S(d)).By using the elementary methods, one of the importantproperties of the formula is exposed and also proved.