Parity of the coefficients of certain eta-quotients, II: The case of even-regular partitions

TL;DR

This paper investigates the parity properties of m-regular partition functions for even m, proposing a conjecture classifying their odd densities and revealing new parity behaviors and congruences, supported by computational and theoretical evidence.

Contribution

It introduces a conjecture classifying the odd density of m-regular partition functions for even m and explores their parity behaviors, including new congruences and relations to multipartition functions.

Findings

For m=2^j m_0, the odd density of b_m is 1/2 if 2^j < m_0.

When 2^j > m_0, b_m is even on infinitely many subprogressions.

b_m often matches the parity of multipartition functions p_t on certain subprogressions.

Abstract

We continue our study of the density of the odd values of eta-quotients, here focusing on the $m$-regular partition functions $b_m$ for $m$ even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let $m = 2^j m_0$ with $m_0$ odd. If $2^j < m_0$, then the odd density of $b_m$ is $1/2$; moreover, such density is equal to $1/2$ on every (nonconstant) subprogression $An+B$. If $2^j > m_0$, then $b_m$, which is already known to have density zero, is identically even on infinitely many non-nested subprogressions. This and all other conjectures of this paper are consistent with our ''master conjecture'' on eta-quotients presented in the previous work. In general, our results on $b_m$ for $m$ even determine behaviors considerably different from the case of $m$ odd. Also interesting, it frequently happens that on subprogressions $An+B$, $b_m$ matches the parity of the multipartition functions $p_t$, for certain values of $t$. We make a suitable use of Ramanujan-Kolberg identities to deduce a large class of such results; as an example, $b_{28}(49n+12) \equiv p_3(7n+2) \pmod{2}$. Additional consequences are several ''almost always congruences'' for various $b_m$, as well as new parity results specifically for $b_{11}$. We wrap up our work with a much simpler proof of the main result of a recent paper by Cherubini-Mercuri, which fully characterized the parity of $b_8$.