Study Of The Smarandache Function And Related Functions

The mean value problems of functions play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory. American-Romanian number theorist Florentin Smarandache introduced hundreds of interesting sequences and arithmetical functions, and presented many problems and conjectures in his life. In 1991, he published a book named 'Only problems, Not solutions!'. He presented 105 unsolved arithmetical problems and conjectures about these functions and sequences in it. Many researchers studied these sequences and functions from this book, and obtained important results.In this dissertation, we study the puoperties of Smarandache function, Smarandache primitive function. Smarandache dual function and Smarandache ceil function, and get some beautiful results. We also study the the mean value properties of the Smarandache pseudo-multiples of 5 number sequence by using elementary and analytic methods, the convergence property of an infinite series involving the inferior and superior factorial part sequences, and give an sufficient condition of the convergence property of the series. By studying the properties of the Smarandache-Riemann zeta sequence, we solve a conjecture posed by Murthy. Besides, we study the mean value property sequence of the complement of hexagon number's part and the hybrid mean value property with the divisor function and Euler function. Last, we study the value distribution properties of the prime divisor number function acting on the multi-group combinatorial numbers C_kn^k.