{
  "id": "2203.07863",
  "version": "v1",
  "published": "2022-03-15T13:12:03.000Z",
  "updated": "2022-03-15T13:12:03.000Z",
  "title": "An asymptotic approximation for the Riemann zeta function revisited",
  "authors": [
    "R B Paris"
  ],
  "comment": "8 pages, 0 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We revisit a representation for the Riemann zeta function $\\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\\bf 4} (1997) 449--470. Use of the uniform asymptotics of the incomplete gamma function produces an asymptotic-like expansion for $\\zeta(s)$ on the critical line $s=1/2+it$ as $t\\to+\\infty$. The main term involves the original Dirichlet series smoothed by a complementary error function of appropriate argument together with a series of correction terms. It is the aim here to present these correction terms in a more user-friendly format by expressing then in inverse powers of $\\omega$, where $\\omega^2=\\pi s/(2i)$, multiplied by coefficients involving trigonometric functions of argument $\\omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-15T13:12:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "33B20",
      "34E05",
      "41A60"
    ],
    "keywords": [
      "riemann zeta function",
      "asymptotic approximation",
      "correction terms",
      "incomplete gamma function produces",
      "complementary error function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}