{
  "id": "1307.6606",
  "version": "v3",
  "published": "2013-07-24T23:14:25.000Z",
  "updated": "2015-08-06T23:15:06.000Z",
  "title": "Counting Square Discriminants",
  "authors": [
    "Thomas A. Hulse",
    "E. Mehmet Kıral",
    "Chan Ieong Kuan",
    "Li-Mei Lim"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by investigating the analytic properties of a certain double Dirichlet series, which is a shifted convolution sum of certain classical automorphic forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-23T01:34:08.000Z",
      "abstract": "Hee Oh and Nimish Shah prove that the number of integral binary quadratic forms whose coefficients are bounded by a quantity $X$, and with discriminant a fixed square integer $d$, is $cX\\log X + O(X(\\log X)^{\\frac 34})$. This result was obtained by the use of ergodic methods. Here we use the method of shifted convolution sums of Fourier coefficients of certain automorphic forms to obtain a sharpened result of a related asymptotic, obtaining a second main term and an error of $O(X^{\\frac 12})$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-08-06T23:15:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "counting square discriminants",
      "integral binary quadratic forms",
      "second main term",
      "fixed square integer",
      "fourier coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6606H"
    }
  }
}