Eta-quotients and divisibility of certain partition functions by powers of primes

TL;DR

This paper investigates the divisibility properties of specific overpartition functions by prime powers, establishing almost always divisibility for primes dividing certain parameters and identifying infinite sets where divisibility fails.

Contribution

It extends divisibility results of overpartition functions to primes greater than or equal to 5 and improves existing theorems on regular partitions and overpartitions.

Findings

Almost always divisible by powers of primes dividing parameters

Existence of infinite n where divisibility does not hold

Improved bounds for divisibility of regular partitions

Abstract

Andrews' $(k, i)$-singular overpartition function $\overline{C}_{k, i}(n)$ counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. In recent times, divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by $2$ and $3$ are studied for certain values of $\ell$. In this article, we study divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by primes $p\geq 5$. For all positive integer $\ell$ and prime divisors $p\geq 5$ of $\ell$, we prove that $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ are almost always divisible by arbitrary powers of $p$. For $s\in \{3, 4, 6\}$, we next show that the set of those $n$ for which $\overline{C}_{s\cdot\ell, \ell}(n) \not\equiv 0\pmod{p_i^k}$ is infinite, where $k$ is a positive integer satisfying $p_i^{k-1}\geq \ell$. We further improve a result of Gordon and Ono on divisibility of $\ell$-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of $\ell$-regular overpartitions by powers of certain primes.