Several Problems On Smarandache Function

As an old filed of mathematics that has continuously developed, number theory mainly studies laws of number, especially properties of integers. The Smarandache function is an important problem of the number theory. As it was constantly studied by people, many types of equations and research methods have appeared.In this paper, some problems about Smarandache functions are mainly explored by methods of elementary and analytic number theories by reading lots of books and literature about the functions. The research content includes equation solvability of Smarandache LCM dual functions, equations of Smarandache LCM dual functions, mean value of Smarandache power functions and solutions of Smarandache second-order even functions. In this paper, several results are obtained as follows:1. The solvability of, two equations concerning Smarandache LCM dual functions, is explored by elementary number theory and separately discussed. All of their specific positive integer solutions are determined.2. Problem about mean estimation for famous Smarandache power functions is mainly examined by elementary and analytic number theories. In other words, the mean between Smarandache power functions and W(n), Smarandache power functions and S(n) is determined based on the sequence of simple number.3. an equation including()kS n(a Smarandache Ceil dual function) and W(n)(a prime factor function), is studied. Then, all positive integer solutions of the equation are determined when k equals to 6.4. Solution of W(an equation about Smarandache second-order dualfunctions) is studied by elementary methods.