{
  "id": "1305.5502",
  "version": "v1",
  "published": "2013-05-23T17:58:33.000Z",
  "updated": "2013-05-23T17:58:33.000Z",
  "title": "On the Probability of Relative Primality in the Gaussian Integers",
  "authors": [
    "Bianca De Sanctis",
    "Samuel Reid"
  ],
  "comment": "6 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\\gcd(z_{1},z_{2})=1) = \\frac{1}{\\zeta_{\\Q(i)}(2)}$$ where $z_{1},z_{2} \\in \\mathbb{Z}[i]$ are randomly chosen and $\\zeta_{\\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theorem to arbitrary number fields that are principal ideal domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-23T17:58:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M20",
      "11K99"
    ],
    "keywords": [
      "gaussian integers",
      "relative primality",
      "probability",
      "dedekind zeta function",
      "arbitrary number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.5502D"
    }
  }
}