Parity of the coefficients of certain eta-quotients

TL;DR

This paper studies the parity of coefficients in eta-quotients, revealing new results on their odd values, lacunarity, and congruences, especially for m-regular partitions up to m=28, and proposes a broad conjecture extending classical partition function conjectures.

Contribution

It introduces new theorems and conjectures on the parity and congruence properties of eta-quotients, extending classical results and exploring their density and self-similarity.

Findings

Established lacunarity modulo 2 for certain coefficients

Discovered self-similarities modulo 2 in eta-quotients

Identified infinite families of congruences in arithmetic progressions

Abstract

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of $m$-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all $m \leq 28$, we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area.