On the density of the odd values of the partition function, II: An infinite conjectural framework

Abstract

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that p(n) is odd exactly 50% of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality.A striking consequence is that, under suitable existence conditions, if any t-multipartition function is odd with positive density and t ≡ 0 ( mod 3), then p(n) is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today.Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.Hence, p t (n) denotes the number of t-multipartitions (also called t-colored partitions) of n.The goal of this paper is to provide additional insight into the long-standing question of estimating the density of the odd values of the partition function. As Paul Monsky once pointedly stated [22], "the best minds of our generation haven't gotten anywhere with understanding the parity of p(n)." It is a famous conjecture of Parkin and Shanks [30]