{
  "id": "1202.1335",
  "version": "v3",
  "published": "2012-02-07T02:36:32.000Z",
  "updated": "2012-04-04T03:35:16.000Z",
  "title": "Representations of analytic functions as infinite products and their application to numerical computations",
  "authors": [
    "Marcin Mazur",
    "Bogdan V. Petrenko"
  ],
  "comment": "Several editorial changes have been made",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $D$ be an open disk of radius $\\le 1$ in $\\mathbb C$, and let $(\\epsilon_n)$ be a sequence of $\\pm 1$. We prove that for every analytic function $f: D \\to \\mathbb C$ without zeros in $D$, there exists a unique sequence $(\\alpha_n)$ of complex numbers such that $f(z) = f(0)\\prod_{n=1}^{\\infty} (1+\\epsilon_nz^n)^{\\alpha_n}$ for every $z \\in D$. From this representation we obtain a numerical method for calculating products of the form $\\prod_{p \\text{prime}} f(1/p)$ provided $f(0)=1$ and $f'(0) = 0$; our method generalizes a well known method of Pieter Moree. We illustrate this method on a constant of Ramanujan $\\pi^{-1/2}\\prod_{p \\text{prime}} \\sqrt{p^2-p}\\ln(p/(p-1))$. From the properties of the exponents $\\alpha_n$, we obtain a proof of the following congruences, which have been the subject of several recent publications motivated by some questions of Arnold: for every $n \\times n$ integral matrix $A$, every prime number $p$, and every positive integer $k$ we have $\\text{tr} A^{p^k} \\equiv \\text{tr} A^{p^{k-1}} (\\text{mod}\\,{p^k})$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-04-04T03:35:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11Y60",
      "30E10",
      "30J99",
      "40A30",
      "40A20",
      "11A07",
      "11C20"
    ],
    "keywords": [
      "analytic function",
      "infinite products",
      "numerical computations",
      "representation",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.1335M"
    }
  }
}