On the Algebraic Reformulation of the Partition Function

A partition of a non-negative integer n is any non-decreasing sequence of integers which sum to n; the partition function, p(n), counts the number of partitions of an integer. The partition function has long enchanted the minds of great mathematicians, dating from Euler's attempts at calculating the value of this function in the 1700's, to Hardy and Ramanujan's asymptotic approach in the early twentieth century, through to Rademacher's representation as an explicit infinite series mid-century.

In the next chapter, we will explore more definitions and examples relating to the partition function, as well as some of the historical methods used to computer the number of partitions. This will include recursive formulas, formulas based on partitions having a specific largest value for each term, and the asymptotic formula.

The third chapter will examine Rademacher's explicit representation of the partition function as an infinite series, including a closer look at the reformulation of a key inner sum from his definition of the partition function. There will also be several examples of computations of this inner sum.

The fourth chapter focuses on Ono and Bringmann's reformulation of the partition function, which is the main result discussed in this thesis. This reformulation takes the explicit representation of p(n) as an infinite series to a finite, closed sum relating to a Poincaré series. This is historically significant in the field of partition theory, as it is the first instance of a finite sum for p(n).

Finally, in the fifth chapter we will discuss a further reformulation of p(n) due to Ono and Bruinier; this additional reformulation allows an expansion of the Poincaré series, providing a computable estimate of p(n) in terms of algebraic numbers. Based on this expansion and certain quadratic forms, the details of two examples of computing p(n) will be included.