{
  "id": "1609.04795",
  "version": "v1",
  "published": "2016-09-10T03:31:50.000Z",
  "updated": "2016-09-10T03:31:50.000Z",
  "title": "Further Exploration of Riemann's Functional Equation",
  "authors": [
    "Michael Milgram"
  ],
  "comment": "17 pages, 5 figures, This paper is an extension of, and should be read in conjunction with, the previous paper arXiv 1510.06333",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization from the critical line to the complex plane. A simpler statement of the relationship that exists between the real and imaginary components of $\\zeta(s)$ and $\\zeta^{\\prime}(s)$ on opposing sides of the critical line is developed, reducing to a simpler statement of the same result on the critical line. An analytic expression is obtained for the sum of the arguments of $\\zeta(s)$ on opposite sides of the critical line, reducing to the analytic expression for $arg(\\zeta(1/2+i\\rho))$ first obtained in the previous work. Relationships are obtained between various combinations of $|\\zeta(s)|$ and $|\\zeta^{\\prime}(s)|$, particularly on the critical line, and it is demonstrated that $arg(\\zeta(1/2+i\\rho))$ and $arg(\\zeta^{\\prime}(1/2+i\\rho))$ uniquely define $|\\zeta(1/2+i\\rho)|$. A comment is made about the utility of such results as they might apply to putative proofs of Riemann's Hypothesis (RH).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-10T03:31:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11M99",
      "26A09",
      "30B40",
      "30E20",
      "30C15",
      "33C47",
      "33B99",
      "33F99"
    ],
    "keywords": [
      "riemanns functional equation",
      "critical line",
      "exploration",
      "complex plane",
      "analytic expression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}