{
  "id": "2101.06900",
  "version": "v1",
  "published": "2021-01-18T06:48:31.000Z",
  "updated": "2021-01-18T06:48:31.000Z",
  "title": "Eta-quotients and divisibility of certain partition functions by powers of primes",
  "authors": [
    "Ajit Singh",
    "Rupam Barman"
  ],
  "comment": "2^4 pages. arXiv admin note: text overlap with arXiv:2009.11602",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-18T06:48:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partition functions",
      "eta-quotients",
      "singular overpartition function",
      "positive integer",
      "arbitrary powers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}