Identical Vanishing of Coefficients in the Series Expansion of Eta Quotients, modulo 4, 9 and 25

TL;DR

This paper investigates when coefficients of eta quotients vanish modulo 4, 9, and 25, revealing specific congruence relations and extending previous results to infinite families and individual cases using modular form theory.

Contribution

It extends the understanding of eta quotient coefficient congruences modulo 4, 9, and 25, including infinite family results and new individual theorems using modular forms.

Findings

For m=4,9, infinite family results established.

Specific congruence relations between eta quotient coefficients.

Experimental results and individual theorems for m=25.

Abstract

Let $A(q)=\sum_{n=0}^{\infty}a_n q^n$ and $B(q)=\sum_{n=0}^{\infty}b_n q^n$ be two eta quotients. Previously, we considered the problem of when \[ a_n=0 <=> b_n=0. \] Here we consider the ``mod $m$'' version of this problem, i.e. eta quotients $A(q)$ and $B(q)$ and integers $m>1$ such that \[ a_n \equiv 0 \pmod m <=> b_n \equiv 0 \pmod m? \] We found results for $m=p^2$, $p=2, 3$ and $5$. For $m=4,9$, we found results which apply to infinite families of eta quotients. For example: Let $A(q)$ have the form \begin{equation} A(q) = f_1^{3j_1+1}\prod_{3\nmid i}f_i^{3j_i}\prod_{3|i}f_i^{j_i} =: \sum_{n=0}^{\infty}a_nq^n,\,\,B(q) = \frac{f_3}{f_1^3}A(q) =: \sum_{n=0}^{\infty}b_nq^n \end{equation} with $f_{k}=\prod_{n=1}^{\infty}(1-q^{kn})$. Then \begin{align*} a_{3n}-b_{3n}&\equiv 0\pmod 9,\\ 2a_{3n+1}+b_{3n+1}&\equiv0\pmod 9,\\ a_{3n+2}+2b_{3n+2}&\equiv0\pmod 9. \end{align*} Some of these theorems also had some combinatorial implications, such as the following: Let $p_2^{(3)}(n)$ denote the number of bipartitions $(\pi_1, \pi_2)$ of $n$ where $\pi_1$ is 3-regular. Then \begin{equation*} p_2^{(3)}(n)\equiv0\pmod 9 <=> n\text{ is not a generalized pentagonal number}. \end{equation*} In the case of $m=25$, we do not have any general theorems that apply to an infinite family of eta quotients. Instead we give two tables of results that appear to hold experimentally. We do prove some individual results (using theory of modular forms), such as the following: Let the sequences $\{c_n\}$ and $\{d_n\}$ be defined by \begin{equation*} f_1^{10}=:\sum_{n=0}^{\infty}c_nq^n, \hspace{25pt} f_1^{5}f_5=:\sum_{n=0}^{\infty}d_nq^n. \end{equation*} Then \begin{equation*} c_n \equiv 0 \pmod{25} <=> d_n \equiv 0 \pmod{25}. \end{equation*}