On the density of the odd values of the partition function

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 $δ_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $δ_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $δ_1>0$; 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 $δ_t=1/2$ 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.