About Smarandache Function And Equations Mean

American Romania number theoretic expert Professor F.Smarandache refers to many special sequences and105unsolved problems of arithmetical function in book《only problems, not solutions》. These problems aroused the enthusiasm of a number of theory experts at home and abroad and of Numerous Theory lovers. They devoted themselves to studying these problems and pushed the rapid development of the theory, leaving an indelible contribution in the history of the development of Numerous Theory.Basing on the interest in the above problems, we discuss the mean value and the equation for Smarandache problem, and get some good conclusions as follows:1. We study the type of δα(n)(S(n)-P(n))2and δα(n)(SM(n)-P(n))2presenting the asymptotic formulas, and obtain two inferences.where s the largest prime factor of n.2. We study the mean value of J(n) on special series and present the asymptotic formulas, where J(n) is the number of all the Dirichlet original features of n3. We discuss the solutions of Smarandache dual function equation and obtain positive integer of the equation.where s*(n)=max{m|m (?) N, m!|n}; Ω(n) is the number of all the element factors of n including the multiplicity; ω(n) is the number of all the different element factors of n.