On the density of the odd values of the partition function

TL;DR

This paper introduces a new approach to studying the distribution of odd values of the partition function, relating densities of multipartition functions and conjecturing their equidistribution modulo 2.

Contribution

It establishes a link between the densities of odd values of multipartition functions and the partition function, proposing that all such functions are likely equidistributed modulo 2.

Findings

If certain multipartition functions have positive density of odd values, then the partition function also has positive density.

Current results do not guarantee the partition function is odd for a significant proportion of n.

Conjecture that all multipartition functions are equally distributed modulo 2 with density 1/2.

Abstract

The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo 2. Our main result will relate the densities, say $\delta_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $\delta_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $\delta_1>0$; that is, $p(n)$ itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that $p(n)$ is odd for $\sqrt{x}$ values of $n\le x$. In general, we conjecture that $\delta_t=1/2$ for all $t$ odd, i.e., that similarly to the case of $p(n)$, all multipartition functions are in fact equidistributed modulo 2. Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu.