{
  "id": "1702.04708",
  "version": "v1",
  "published": "2017-02-15T18:36:31.000Z",
  "updated": "2017-02-15T18:36:31.000Z",
  "title": "On correlations between class numbers of imaginary quadratic fields",
  "authors": [
    "Vinay Kumaraswamy"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $h(-n)$ be the class number of the imaginary quadratic field with discriminant $-n$. We establish an asymtotic formula for correlations involving $h(-n)$ and $h(-n-l)$, over fundamental discriminants that avoid the congruence class $1\\pmod{8}$. Our result is uniform in the shift $l$, and the proof uses an identity of Gauss relating $h(-n)$ to representations of integers as sums of three squares. We also prove analogous results on correlations involving $r_Q(n)$, the number of representations of an integer $n$ by an integral positive definite quadratic form $Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-15T18:36:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "imaginary quadratic field",
      "class number",
      "correlations",
      "integral positive definite quadratic form",
      "congruence class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}