Related To The Nature Of The Smarandache Function And Sequence

Number theory has always been called the royal crown of mathematics. It is the core content of analytic number theory to make researches on the properties of all kinds of arithmetical functions and sequences. Florentin Smarandache, who is a famous Romanian professor of number theory, gave many new arith-metical functions and sequences, and then put forward some related problems about these functions and sequences. What he did provides a direction for the researchers on number theory. After the professor gave the arithmetical func-tions Smarandache S(n), many scholars made researches on it and got a number of valuable conclusions, which promote a further development of number theory. However many problems which the professor put forward have been unsolved, which may bring a new time for the development of number theory in the future.Because of the above reasons, the dissertation chooses several unsolved ques-tions from the problems that Florentin Smarandache put forward, and then makes use of elementary and analytic ways to study them. Specifically the three questions are as follows:(1) an equation about the arithmetic functions SL(n) and SM(n):(2) a distribution question of mean value about the Pseudo-Smarandache function Z(n); (3) the properties of the permutation sequence PM(n). At last, the dissertation get the following results:1. The solvability of an equation about the arithmetic functions SL(n) and SM(n) is studied, and all the positive integer solutions of the equation are given. At the same time a Dirichlet series is put forward, the astringency of the series is proved , and the distribution formula of the arithmetical function (?) on the solution gather of the equation is gotten.2. The distribution property of the Pseudo-Smarandache function Z(n) is studied, and a asymptotic formula of (?) Z(n)A(n) is given by using the analytical method at last.3. For any positive integer n, the permutation sequence PM(n) is defined as follow:PM(n)= 135…(2n-1)(2n)…42. According to the research for the elements of the sequence, a conjecture is put forward and proved. According to the above conclusion, the factor form of the elements is studied, and then a related theorem is gotten.