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

William J. Keith, Fabrizio Zanello

Published 2023-02-01Version 1

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 $b_8$.