Identical Vanishing of Coefficients in the Series Expansion of Eta Quotients, modulo 4, 9 and 25
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866913962037084160 |
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| author | Huber, Tim McLaughlin, James Ye, Dongxi |
| author_facet | Huber, Tim McLaughlin, James Ye, Dongxi |
| contents | 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 $(π_1, π_2)$ of $n$ where $π_1$ 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*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Identical Vanishing of Coefficients in the Series Expansion of Eta Quotients, modulo 4, 9 and 25 Huber, Tim McLaughlin, James Ye, Dongxi Number Theory 11F33 (Primary) 11B65, 11F11 (Secondary) 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 $(π_1, π_2)$ of $n$ where $π_1$ 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*} |
| title | Identical Vanishing of Coefficients in the Series Expansion of Eta Quotients, modulo 4, 9 and 25 |
| topic | Number Theory 11F33 (Primary) 11B65, 11F11 (Secondary) |
| url | https://arxiv.org/abs/2507.20298 |