Deducing the positive odd density of $p(n)$ from that of a multipartition function: An unconditional 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 $A\equiv \pm 1(\mathrm{mod}6)$ 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 $A\equiv 3(\mathrm{mod}6)$ 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.

Minor updates. To appear in the J. of Number Theory