{
  "id": "2108.03932",
  "version": "v1",
  "published": "2021-08-09T10:51:05.000Z",
  "updated": "2021-08-09T10:51:05.000Z",
  "title": "Asymptotic Formulas and Sign-changes of Coefficients of Hauptmoduls on $Γ_0(p)$",
  "authors": [
    "Liuquan Wang"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We study the coefficients of the hauptmodul $(\\eta(\\tau)/\\eta(p\\tau))^{24/(p-1)}$ on $\\Gamma_0(p)$ with $p\\in \\{2,3,5,7,13\\}$, where $\\eta(\\tau)$ is the Dedekind eta function. We provide asymptotic formulas for its coefficients and give explicit characterizations of its signs. For the cases $p=2$ or $3$, we also give alternative proof for its sign-change properties without using the asymptotic formulas.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-09T10:51:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D57",
      "11E25",
      "11F27",
      "11P83",
      "11P84"
    ],
    "keywords": [
      "asymptotic formulas",
      "coefficients",
      "hauptmodul",
      "dedekind eta function",
      "explicit characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}