{
  "id": "1206.1613",
  "version": "v1",
  "published": "2012-06-07T20:56:27.000Z",
  "updated": "2012-06-07T20:56:27.000Z",
  "title": "Average Number of Lattice Points in a Disk",
  "authors": [
    "Sujay Jayakar",
    "Robert S. Strichartz"
  ],
  "comment": "11 pages, 12 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "The difference between the number of lattice points in a disk of radius $\\sqrt{t}/2\\pi$ and the area of the disk $t/4\\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \\leq t \\leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-07T20:56:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "42B99"
    ],
    "keywords": [
      "lattice points",
      "average number",
      "eigenvalue counting function",
      "weyl asymptotic estimate",
      "standard flat torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1613J"
    }
  }
}