{
  "id": "1710.06834",
  "version": "v1",
  "published": "2017-10-18T17:28:33.000Z",
  "updated": "2017-10-18T17:28:33.000Z",
  "title": "Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture",
  "authors": [
    "Daniel Fiorilli",
    "James Parks",
    "Anders Södergren"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order terms when the support of the Fourier transform of the corresponding test function reaches the point $1$. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-18T17:28:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11M50"
    ],
    "keywords": [
      "quadratic dirichlet",
      "low-lying zeros",
      "transition",
      "functions ratios conjecture",
      "corresponding test function reaches"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}