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

TL;DR

This paper develops an infinite framework of conjectural identities modulo 2 related to the parity of the partition function, aiming to understand when p(n) is odd with positive density, but the main conjecture remains unproven.

Contribution

It introduces a doubly-indexed, infinite set of conjectural identities modulo 2 and discusses their potential to prove the parity conjecture for the partition function.

Findings

Framework extends previous work on partition parity.

Conditional results relate odd density of multipartition functions to p(n).

Employs algebraic and analytic methods, including modular form techniques.

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 $t\not \equiv 0$ (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.