{
  "id": "1510.06333",
  "version": "v1",
  "published": "2015-10-20T18:59:25.000Z",
  "updated": "2015-10-20T18:59:25.000Z",
  "title": "Exploring Riemann's Functional Equation",
  "authors": [
    "Michael Milgram"
  ],
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "An equivalent, but variant form of the Riemann functional equation is explored, and several discoveries are made. Properties of the Riemann zeta function $\\zeta(s)$ from which a necessary and sufficient condition for the existence of zeros in the critical strip are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for $arg(\\zeta(s))$ on the critical line $s=1/2+i\\rho$, the consequences of which are explored, and the \"LogZeta\" function is introduced. A singular linear transform between the real and imaginary components of $\\zeta$ and $\\zeta^\\prime$ on the critical line is derived, and an implicit relationship for locating a zero ($\\rho=\\rho_0$) on the critical line is found between the arguments of $\\zeta(1/2+i\\rho)$ and $\\zeta^{\\prime}(1/2+i\\rho)$. Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent is analytically evaluated and verified to be equivalent to RH as claimed, but RH is not proven. It is proven that the order of a zero on the critical line is never even, and that the derivative $\\zeta^{\\prime}(1/2+i\\rho)$ will never vanish on the punctured critical line ($\\rho\\neq\\rho_0$), nor at a simple zero. Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of $\\zeta(s)$ on the critical line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-20T18:59:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11M99",
      "26A09",
      "30B40",
      "30E20",
      "30C15",
      "33C47",
      "33B99",
      "33F99"
    ],
    "keywords": [
      "exploring riemanns functional equation",
      "critical line",
      "equivalent",
      "riemann functional equation",
      "riemann zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151006333M"
    }
  }
}