{
  "id": "2406.07905",
  "version": "v1",
  "published": "2024-06-12T06:15:44.000Z",
  "updated": "2024-06-12T06:15:44.000Z",
  "title": "Arithmetic density and congruences of $\\ell$-regular bipartitions $II$",
  "authors": [
    "Nabin Kumar Meher"
  ],
  "comment": "Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2406.06224",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $ B_{\\ell}(n)$ denote the number of $\\ell-$regular bipartitions of $n.$ In 2013, Lin \\cite{Lin2013} proved a density result for $B_4(n).$ He showed that for any positive integer $k,$ $B_4(n)$ is almost always divisible by $2^k.$ In this article, we improved his result. We prove that $B_{2^{\\alpha}m}(n)$ and $B_{3^{\\alpha}m}(n)$ are almost always divisible by arbitrary power of $2$ and $3$ respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for $B_2(n)$ and $B_4(n)$ by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on nilpotency of Hecke operator, we also find an infinite families of congruences modulo arbitrary power of $2$ satisfied by $B_{2^{\\alpha}}(n).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-12T06:15:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "regular bipartitions",
      "arithmetic density",
      "congruences modulo arbitrary power",
      "hecke eigenform theory",
      "infinities families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}