The Equation Solver Of Arithmetic Functions With The Mean Estimate

Florentin Smarandache from Romania and R.K.Guy from Canada, who are experts on number theory, propounded some unsolved questions on number theory in "Only Problems, Not Solutions!" and "some unsolved questions on number theory", aroused people's enthusiasm for the study of number theory. Some of these issues have been resolved satisfactorily by a number of foreign experts on number theory, and get a number of results of important theoretical value, but there are many problems to be solved in the future.In this paper, based on learning and exploration about Smarandache prob-lem, we think simply about issues of Smarandache function by elementary and analytical methods. We give a mean value estimate on Smarandache function, solve an equation on the Smarandache power function, and introduce a new arithmetic function. The content include three aspects:1. We obtain a sharper asymptotic formula by studying SL(n) function. That is, for any real number x> 1, we have the asymptotic formula2. For any positive integer n, the equation holds if and only if n= 1,2,3.3. We define a new function g(n), and prove there are a number of positive integer groups (m1, m2,…,mk) satisfying inequality For any positive integer k≥3, we prove there are infinite positive integers groups (m1,m2,…, mk) satisfying the equation...