{
  "id": "1601.06833",
  "version": "v1",
  "published": "2016-01-25T22:16:43.000Z",
  "updated": "2016-01-25T22:16:43.000Z",
  "title": "Low-lying zeros of quadratic Dirichlet $L$-functions: Lower order terms for extended support",
  "authors": [
    "Daniel Fiorilli",
    "James Parks",
    "Anders Södergren"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\\phi$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\\sigma$ of the support of $\\hat \\phi$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\\sigma=1$ involves the quantity $\\hat \\phi (1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-25T22:16:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11M50",
      "11L40"
    ],
    "keywords": [
      "low-lying zeros",
      "quadratic dirichlet",
      "extended support",
      "function field case",
      "real primitive characters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160106833F"
    }
  }
}