{
  "id": "1211.6725",
  "version": "v2",
  "published": "2012-11-28T20:08:15.000Z",
  "updated": "2013-02-14T19:59:19.000Z",
  "title": "Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve",
  "authors": [
    "Vorrapan Chandee",
    "Yoonbok Lee",
    "Sheng-chi Liu",
    "Maksym Radziwiłł"
  ],
  "comment": "This work was initiated during the Arithmetic Statistics MRC program at Snowbird, Utah. Corollary 3 and Section 7 are added",
  "categories": [
    "math.NT"
  ],
  "abstract": "Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet $L$-functions are simple. This improves on earlier work of \\\"{O}zl\\\"{u}k which gives a proportion of at most 86%. We further compute an $q$-analogue of the Pair Correlation Function $F(\\alpha)$ averaged over all primitive Dirichlet $L$-functions in the range $|\\alpha| < 2$ . Previously such a result was available only when the average included all the characters $\\chi$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-14T19:59:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26"
    ],
    "keywords": [
      "asymptotic large sieve",
      "primitive dirichlet",
      "simple zeros",
      "pair correlation function",
      "generalized riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.6725C"
    }
  }
}