On The Smarandache Function And A Class Of Dedekind Sums Nature Study

The properties of arithmetical functions and special sequences are always the kernel of the study in number theory. In 1993, Florentin Smarandache, who is a famous American-Romanian number theorist, published a book named "Only problem, Not solutions!" in Xiquan Publishing House in America. In this book, he proposed 105 unsolved problems and conjectures about arithmetical functions and special sequences. Then many scholars made intensive studies of them, and obtained some important valued results on theory. Besides these, because of the importance of Dedekind sum and the extensive applications, the study of this sum is a focus in number theory these days. Many researchers defined various sums analogous to Dedekind sum, and made worthy and interesting study on them.Based on the interests in above problems, we make use of elementary and analytic methods to study the Smarandache function and the Smarandache dual function. We also define a sum analogous to Dedekind sum and study some properties of it. Specifically, the following results are obtained:1. Using elementary methods and Pythagorean triples, the solvability of a special equation involving the Smarandache function is discussed and necessary conditions for the equation having solutions are given. At the same time, a problem proposed by M. Bencze is solved.2. Let n be any positive integer,(?)(n) and S*(n) be the Euler totient and the Smarandache dual function, the solvability of the equation (?) S*(d)= (?)(n) is studied by elementary methods. 3. A sum analogous to Dedekind sum, is introduced and called likewise Dedekind sum. where (?)Rp denotes a is the quadratic residue mod p. The relation between D(h.p) and Cochrane sum and the related order estimation are given by using elementary and analytic methods.