{
  "id": "1111.1950",
  "version": "v1",
  "published": "2011-11-07T12:09:50.000Z",
  "updated": "2011-11-07T12:09:50.000Z",
  "title": "Exact & Numerical Tests of Generalised Root Identities for non-integer μ",
  "authors": [
    "Richard Stone"
  ],
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We consider the generalised root identities introduced in [1] for simple functions, and also for \\Gamma(z+1) and \\zeta(s). In this paper, unlike [1], we focus on the case of noninteger \\mu. For the simplest function f(z)=z, and hence for arbitrary polynomials, we show that they are satisfied for arbitrary real {\\mu} (and hence for arbitrary complex {\\mu} by analytic continuation). Using this, we then develop an asymptotic formula for the derivative side of the root identities for \\Gamma(z+1) at arbitrary real \\mu, from which we are able to demonstrate numerically that \\Gamma(z+1) also satisfies the generalised root identities for arbitrary \\mu, not just integer values. Finally we examine the generalised root identites for {\\zeta} also for non-integer values of \\mu. Having shown in [1] that {\\zeta} satisfies these identities exactly for integer \\mu>1 (and also for \\mu=1 after removal of an obstruction), in this paper we present strong numerical evidence first that {\\zeta} satisfies them for arbitrary \\mu>1 where the root side is classically convergent, and then that this continues to be true also for -1<\\mu<1 where C\\'esaro divergences must be removed and C\\'esaro averaging of the residual partial-sum functions is required (when \\mu<0). Careful consideration of a neighbourhood of \\mu=0 also sheds light on the appearance of the 2d ln-divergence that was handled heuristically in [1] and why the assignment of 2d C\\'esaro limit 0 to this in [1] is justified. The numerical calculations for \\mu>0 are bundled in portable R-code; the code for the case -1<\\mu<0, including the C\\'esaro averaging required when \\mu<0, is in VBA. Both the R-scripts and XL spreadsheet are made available with this paper, along with supporting files, and can be readily used to further verify these claims.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-07T12:09:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalised root identities",
      "numerical tests",
      "non-integer",
      "arbitrary real",
      "strong numerical evidence first"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.1950S"
    }
  }
}