Bhargava Factorials

The motivation for this project came from The Factorial Function and Gener- alizations, a paper written by the Fields Medal winning mathematician Manjul Bhargava. Bhargava wrote the paper as his thesis at Harvard University. The paper introduces generalizations of the factorial function, they are called the \generalized factorials" or known by the author's name as \Bhargava factorials". In this paper we study the notion of P-ordering, rst introduced by Bhargava in 1996, of an arbitrary subset S of the ring Z: We revisit some of the applications of usual factorial function in number theory. Then when we get to the generalized factorial of any subset S of the ring Z, we check if they have the same application and it is astounding to nd out that the generalized factorial function of any subset S of Z plays the same role as the usual factorial function of a non-negative integer in number theory. The structure of this thesis and Bhargava's aforementioned paper are similar. The topics covered here are picked up from the ones that Bhargava has discussed in his paper. It will seem like a repetition of the paper, The Factorial Function and Generalizations, though the main purpose of this research is to explain Bhargava's work on generalized factorials and give more detail where I felt it was necessary