The Mean Equation Of The Smarandache Function And Other Related Issues

Number theory takes a special status in mathematics, Gauss has praised: "Number theory is the royal crown of mathematics." And all knows, number theory is a discipline of studying the arithmetical function's properties, many famous number theory problems are closely connected with them.Professor Florentin Smarandache is American-Romanian, who is a well-known scholar in number theory. He wrote a book "Only problems. Not-solutions" in 1993, which presented 105 unsolved number theory problems. It took scholars'attention in the academic research. With tireless efforts, scholars have got a lot of significance research results, which made a further development of number theory. However, in the past eighteen years, all the questions of professor F.Smarandache have not been solved completely, but the research in the past eighteen years made the question that had not been solved still be more charm, which urged the more researcher to explore,study.However, all the questions of professor F.Smarandache have not yet fully resolved, while may make great development of number theory. Hence, in this paper, we resolved a few unresolved issues in "Only problems. Not-solutions," analyzed and studied on them by elementary methods in analytic number theory and elementary number theory, specifically, the main problems are in the following:1. By elementary number theory, we study positive integer solutions on the Smarandache multiplicative function SM(n) and pseudo-Smarandache Z(n) equations, and give specific solution sets.2. By elementary and analytic method, we study the mean value problem of the pseudo-Smarandache Z* (n), and give an asymptotic formula. 3. By analytical method, we study new properties of the Cochrane sequence and some new hybrid mean about the Cochrane sum, and get several interesting asymptotic formulas.4. By elementary number theory method, we study the properties of the near pseudo-Smarandache function Ut(n) and obtained two interesting identities about it, and give a general method for solving the identity.