{
  "id": "2003.10616",
  "version": "v1",
  "published": "2020-03-24T02:02:34.000Z",
  "updated": "2020-03-24T02:02:34.000Z",
  "title": "Rational Approximations via Hankel Determinants",
  "authors": [
    "Timothy Ferguson"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Define the monomials $e_n(x) := x^n$ and let $L$ be a linear functional. In this paper we describe a method which, under specified conditions, produces approximations for the value $L(e_0 )$ in terms of Hankel determinants constructed from the values $L(e_1 )$, $L(e_2 )$, . . . . Many constants of mathematical interest can be expressed as the values of integrals. Examples include the Euler-Mascheroni constant $\\gamma$, the Euler-Gompertz constant $\\delta$, and the Riemann-zeta constants $\\zeta(k)$ for $k \\ge 2$. In many cases we can use the integral representation for the constant to construct a linear functional for which $L(e_0)$ equals the given constant and $L(e_1)$, $L(e_2)$, . . . are rational numbers. In this case, under the specified conditions, we obtain rational approximations for our constant. In particular, we execute this procedure for the previously mentioned constants $\\gamma$, $\\delta$, and $\\zeta(k)$. We note that our approximations are not strong enough to study the arithmetic properties of these constants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-24T02:02:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J17",
      "11C20",
      "15B05"
    ],
    "keywords": [
      "hankel determinants",
      "rational approximations",
      "linear functional",
      "specified conditions",
      "rational numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}