Deducing the positive odd density of $p(n)$ from that of a multipartition function: An unconditional proof

TL;DR

This paper proves unconditionally that the density of odd values of the partition function p(n) is positive, using multipartition functions and recent breakthroughs, advancing understanding of a long-standing conjecture.

Contribution

It provides an unconditional proof linking the odd density of p(n) to that of multipartition functions, building on previous conjectures and recent mathematical breakthroughs.

Findings

Proved the positive odd density of p(n) unconditionally.

Connected the odd density of p(n) to multipartition functions.

Utilized recent advances by Chen in the proof.

Abstract

A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density $1/2$. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any $A \equiv \pm 1\ (\bmod 6)$ the multipartition function $p_A(n)$ has positive odd density, then so does $p(n)$. Similarly, the positive odd density of any $p_{A}(n)$ with $A\equiv 3\ (\bmod 6)$ would imply that of $p_3(n)$. Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture.