{
  "id": "1210.7357",
  "version": "v1",
  "published": "2012-10-27T18:42:58.000Z",
  "updated": "2012-10-27T18:42:58.000Z",
  "title": "A Finite Reflection Formula For A Polynomial Approximation To The Riemann Zeta Function",
  "authors": [
    "Stephen Crowley"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x) = floor(1/x)*(-1+x*floor(1/x)+x) multiplied by s((s+1)/(s-1)). A finite-sum approximation to \\zeta (s) denoted by \\zeta_w(N;s) which has real roots at s=-1 and s=0 is examined and an associated function \\chi (N ; s) is found which solves the reflection formula \\zeta_w (N ; 1 - s) = \\chi (N ; s) \\zeta_w (N ; s). A closed-form expression for the integral of \\zeta_w (N ; s) over the interval s=-1..0 is given. The function \\chi (N ; s) is singular at s=0 and the residue at this point changes sign from negative to positive between the values of N=176 and N=177. Some rather elegant graphs of \\zeta_w(N ; s) and the reflection functions \\chi (N ; s) are also provided. The values \\zeta_w (N ; 1 - n) for integer values of n are found to be related to the Bernoulli numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-27T18:42:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann zeta function",
      "finite reflection formula",
      "polynomial approximation",
      "unit interval map",
      "point changes sign"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.7357C"
    }
  }
}