Parity of the partition function in quadratic progressions

Abstract

The parity of the partition function $p(n)$ is one of the most stubborn problems in number theory. In 2010, the first author conjectured, for square-free $1<D\equiv 23(\mathrm{mod}24)$, that the values $p\left(\frac{D{m}^2+1}{24}\right)$, as $m$ ranges over positive integers with $(m,6)=1$, include infinitely many even and infinitely many odd terms. We prove this conjecture. The key new idea is geometric. Logarithmic derivatives of twisted Borcherds products built from Ramanujan's third-order mock theta functions recast the problem in terms of CM points of discriminant $-D$ on $X_0(6)$. The uniqueness of canonical lifts from characteristic $2$ to characteristic zero shows that the CM points supporting the poles remain distinct after reduction modulo $2$. This fact, combined with an Eisenstein series comparison and a Galois representation argument, rules out constant parity and gives infinitely many values of each parity. The method applies to the coefficients of analogous generalized twisted Borcherds products. These results imply that \[\#\{0\leq n\leq X: p(n)\text{ is odd}\}\gg\sqrt X,\] which is now the best known lower bound for odd values of $p(n)$. The algebraic identities at the heart of this paper, as well as the improved lower bound for odd partition numbers, have been formalized in Lean by AxiomProver.

Corollary 2 is new: New best known lower bound for the appearance of odd values of p(n). This corollary was brought to our attention by Q.-Y. Zheng. His proof (also formalized in Lean) is in the new Appendix