{
  "annotation_id": "5ad09866-3ad6-4a69-a50e-5c185b249d70",
  "date_created": "2026-02-17T05:53:12.435000Z",
  "date_modified": "2026-02-17T05:53:12.435000Z",
  "file_hash": "a4a2db6434c9a1d4965aa89062a6d9b230b380227e89b574604e99479ae26835",
  "private": false,
  "record": {
    "abstract": "The d\u0027Arcais polynomials $P_n(z)$ for $n\\in\\{0,1,\\dots\\}$ are defined as $\\sum_{n=0}^{\\infty} P_n(z) q^n = \\exp(-z\\ln((q;q)_{\\infty}))$ where the $q$-Pochhammer symbol is $(q;q)_{\\infty} = \\prod_{k=1}^{\\infty} (1-q^k)$ for $|q|\u003c1$. Denoting the coefficients for $n \\in \\mathbb{N}$ by the formula $P_n(z) = \\sum_{k=1}^{n} A(2,n,k) z^k/n!$, we prove that $k_n! A(2,n,k_n)/n!$ satisfies a Bahadur-Rao type large deviation formula in the limit $n \\to \\infty$ with $k_n/n \\to \\kappa \\in [0,1)$ as long as $k_n \\to \\infty$. The large deviation rate function is the Legendre-Fenchel transform $g^*(-\\kappa)$ where $g(\\kappa) = f^{-1}(\\kappa)$ for the function $f : (0,\\infty) \\to \\mathbb{R}$ given by $f(y)= \\ln(-\\ln((e^{-y};e^{-y})_{\\infty}))$. We relate this fact to information about the abundancy index.",
    "arxiv_id": "2601.07103",
    "authors": [
      "Shannon Starr"
    ],
    "categories": [
      "math.PR",
      "math.CO",
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Large Deviations for the d\u0027Arcais Numbers",
    "url": "https://arxiv.org/abs/2601.07103",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "9cafa05d-d8ea-46d3-93b6-7d0995e6152b",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}