{
  "id": "2306.00460",
  "version": "v1",
  "published": "2023-06-01T09:03:09.000Z",
  "updated": "2023-06-01T09:03:09.000Z",
  "title": "Spirals of Riemann's Zeta-Function --Curvature, Denseness, and Universality--",
  "authors": [
    "Athanasios Sourmelidis",
    "Jörn Steuding"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "This article deals with applications of Voronin's universality theorem for the Riemann zeta-function $\\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\\zeta(\\sigma+it)$ for real $t$ where $\\sigma\\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $\\zeta(\\sigma+it)$ when $\\sigma>1/2$ and we show that there is a connection with the zeros of $\\zeta'(\\sigma+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-01T09:03:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06"
    ],
    "keywords": [
      "riemanns zeta-function",
      "plane smooth curve appears",
      "voronins universality theorem",
      "small error",
      "imaginary part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}