{
  "id": "2101.07402",
  "version": "v1",
  "published": "2021-01-19T01:40:51.000Z",
  "updated": "2021-01-19T01:40:51.000Z",
  "title": "Dirichlet series for complex powers of the Riemann zeta function",
  "authors": [
    "Winston Alarcon-Athens"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In order to obtain the Dirichlet series for the exponential $\\zeta(s)^z$, we define the notion of multiplicative architecture applicable to every integer $ n > 1 $, to which we associate the concept of degree of $n$. These notions are suitable for defining and studying a sequence $(\\alpha_n(z))_{n \\in {\\Bbb Z}^+}$ of polynomials in the indeterminate $z$ which, being used as coefficients of the respective terms of the Dirichlet series of $\\zeta(s)$ in the semi-plane $\\Re(s)>1$, they convert it into an analitic function $\\varphi_s(z)$ that has the characteristic property of exponential functions: $\\varphi_s(z_1)\\,\\varphi_s(z_2) = \\varphi_s(z_1 + z_2)$, for all $z_1$, $z_2 \\in \\Bbb C$. This result allows us to write $\\zeta(s)^z = \\sum_ {n = 1}^\\infty {\\alpha_n(z) \\over n^s}$ for all $z \\in \\Bbb C$, since $\\varphi_s(1) = \\zeta(s) \\neq 0$ for all $s\\in \\Bbb C$ with $\\Re(s) > 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-19T01:40:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41"
    ],
    "keywords": [
      "dirichlet series",
      "riemann zeta function",
      "complex powers",
      "analitic function",
      "characteristic property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}