{
  "id": "2111.04183",
  "version": "v4",
  "published": "2021-11-07T21:36:24.000Z",
  "updated": "2022-08-29T11:02:04.000Z",
  "title": "Asymptotics for the twisted eta-product and applications to sign changes in partitions",
  "authors": [
    "Walter Bridges",
    "Johann Franke",
    "Taylor Garnowski"
  ],
  "comment": "Typos and grammatical errors fixed",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We prove asymptotic formulas for the complex coefficients of $(\\zeta q;q)_\\infty^{-1}$, where $\\zeta$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions of $n$ with largest part congruent $a$ modulo $b$. Our results imply that, as $n \\to \\infty$, the difference $p(a_1,b,n)-p(a_2,b,n)$ for $a_1 \\neq a_2$ oscillates like a cosine, when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of $\\{p(a,b,n)\\}_{1 \\leq a \\leq b}$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2022-08-29T11:02:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sign changes",
      "twisted eta-product",
      "asymptotic formulas",
      "applications",
      "determine secondary terms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}