{
  "id": "1901.01256",
  "version": "v1",
  "published": "2019-01-06T00:38:34.000Z",
  "updated": "2019-01-06T00:38:34.000Z",
  "title": "An Integral Equation for Riemann's Zeta Function and its Approximate Solution",
  "authors": [
    "Michael Milgram"
  ],
  "comment": "28 pages, 11 Figures, 4 Appendices",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Two obscure identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\\xi(s)$ function, and thus $\\zeta(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates $\\zeta(s)$ anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, I obtain both an analytic expression for $\\zeta(\\sigma+i\\rho)$ everywhere inside the asymptotic ($\\rho\\rightarrow\\infty)$ critical strip, and an approximate solution, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of $\\zeta(\\sigma+i\\rho)$ for different values of $\\sigma$ and equal values of $\\rho$; this is illustrated in a number of Figures. Finally, arguments are presented to show that that the proffered \"approximate\" solution is less approximate and more rigorous than meets the eye.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-06T00:38:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11M99",
      "26A09",
      "30B40",
      "30E20",
      "30C15",
      "33C47",
      "33B99",
      "33F99"
    ],
    "keywords": [
      "riemanns zeta function",
      "integral equation",
      "critical strip",
      "approximate solution predicts",
      "derivations flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}