Parity of the coefficients of certain eta-quotients, III: two special classes

TL;DR

This paper advances the parity study of eta-quotients by examining two classes: the eta-quotients $G_t = \frac{f_t^3}{f_1}$ and pure eta-powers $f_1^t$. It proves instances where the odd-density of coefficients vanishes and constructs numerous even-progressions, connecting these parity phenomena to densities $\delta^{(t)}$ and $\delta^{[t]}$ and to broader conjectures about lacunarity and modular-form structure. The authors develop both general theorems and explicit congruences, using CMSZ lacunarity, modular-form techniques, and quadratic-residue analyses to relate eta-quotients to regular partitions and to derive new identities and corollaries for $c_t(n)$ and $g_t(n)$. They also propose a comprehensive master conjecture on the parity of eta-quotients, detailing expected dichotomous density behavior and posing open problems about prime-specific $p^2$-evenness and progression-free phenomena. Collectively, the work links eta-quotient parity to density results, modular forms, and the structure of partitions, with implications for the Parkin-Shanks parity conjecture and related $q$-series questions.

Abstract

We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on two classes. One consists of eta-quotients of the form $f_t^3/f_1$, a distinguished case of Andrews' singular overpartitions that has recently attracted attention among researchers. In addition, we investigate the parity of certain pure eta-powers $f_1^t$, appending new results to known density theorems.

Theorems & Definitions (53)

• Theorem 1: Zanello, Theorem 4
• Remark 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Remark 9
• Theorem 10