Arithmetic Properties Of The Smarandache Function And Its Related Sequences Study

The study on the properties of arithmetical sequences is the kernel of the research in number theory. In 1993, American-Romanian number theorist Florentin Smarandache published a book named " Only problems, Not solutions!" . In this book, he presented 105 unsolved arithmetical problems and conjectures about special sequences and arithmetical functions. Many researchers studied these sequences and functions from this book, and obtained some important valued results on theory.Based on our interests in the above problems, we use elementary methods and analytic methods to study the arithmetical properties of Smarandache function and its related functions from the following three aspects: (1) The mean value distribution properties of these sequences; (2) Convergence and divergence estimates of the infinity series involving these sequences; (3) The solvability of some equations involving these sequences. Specially, the main achievements contained in this dissertation are as follows:1. Using elementary methods to study the solvability of two equations involving pseuo-Smanrandache function and Euler totient function, give the necessary conditions of these two equations having solutions and the exact solutions in some special cases. Meanwhile, solve the two problems proposed by Chares Ashbacher completely.2. Give the definition of the Smarandache LCM dual function SL^*(n). Using the elementary method to study the calculating problem of a Dirichlet series involving the Smarandache LCM dual function SL^*(n) and the mean value distribution property of SL^*(n), obtain an exact calculating formula and a sharper asymptotic formula for it. Simultaneously we use the elementary methods to study the solutions of two equations involving the Smarandache LCM dual function SL^*(n), and give their all positive solutions.3. Using the elementary method to study the solutions of the equation SP(n^k) =Φ(n), and give its all positive integer solutions for k = 1, 2, 3.4. Using the elementary method to study the solutions of the equationΦ(Φ(n)) = 2^Ω(n), whereΩ(n) defined as follows:Ω(1) = 0; If n > 1 and n =p₁^α1·p₂^α2·····p_k^αk be the prime powers factorization of n, thenΩ(n) = (?)Φ(n) denotes the Euler-totient function, and give all positive integer solutions. Namely, the problem proposed by Dr.Zhang Tianping is solved.