| Number theory, which is called "the queen of mathematics", has always been pursued by many people, and the properties of all kinds of arithmetical functions are always the main subject in number theory. In 1993, a series of new arithmetical functions, sequences and conjectures were presented by Florentin Smarandache in the book《Only Problems, Not Solutions!》, who is a famous American-Romanian number theorist, and he suggested making a study on the relevant problems, which open up a new subject for number theory. Recently, many number theorists and scholars at home and abroad made researches on it and obtained many abundant and excellent results. Meanwhile, many new problems about Smarandache functions arose.Based on the interesting in above Smarandache problems, with the basic "material" prime, we made researches on the arithmetical properties of Smaran-dache functions by elementary and analytical methods. We studied the solv-ability of special equations involving the Smarandache functions perfectly and gave a mean value estimate on Smarandache function. Specifically, the content include the following three aspects:1. Using elementary method to study the solvability of equation involving the Smarandache dual function S* (n) and Smarandache ceil function Sk(n), we proved there are infinite positive integer solutions and gave its concrete forms.2. Combined a new Smarandache function U(n) and its dual function V(n), the distribution properties of(U(n) - V(n))2 is studied by analytic method, and we gave the following interesting asymptotic formulawhereζ(n) is the Riemann zeta-function, and ci (i=2,3,…,k) are computable constants.3. Explored the arithmetical properties of Smarandache double factorial function, structured three equations about the Smarandache double factorial function and used the method of classification and congruence, the solvability of this three equations are solved completely, whereφ(n) is the Euler function, Z(n) is the Pseudo Smarandache function. |
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