{
  "id": "1805.08811",
  "version": "v1",
  "published": "2018-05-22T18:41:44.000Z",
  "updated": "2018-05-22T18:41:44.000Z",
  "title": "Some multidimensional integrals in number theory and connections with the Painlevé V equation",
  "authors": [
    "Estelle Basor",
    "Fan Ge",
    "Michael O. Rubinstein"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study piecewise polynomial functions $\\gamma_k(c)$ that appear in the asymptotics of averages of the divisor sum in short intervals. Specifically, we express these polynomials as the inverse Fourier transform of a Hankel determinant that satisfies a Painlev\\'e V equation. We prove that $\\gamma_k(c)$ is very smooth at its transition points, and also determine the asymptotics of $\\gamma_k(c)$ in a large neighbourhood of $k=c/2$. Finally, we consider the coefficients that appear in the asymptotics of elliptic Aliquot cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-22T18:41:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "33E17"
    ],
    "keywords": [
      "number theory",
      "multidimensional integrals",
      "connections",
      "study piecewise polynomial functions",
      "inverse fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}