{
  "id": "1405.5486",
  "version": "v4",
  "published": "2014-05-21T17:35:10.000Z",
  "updated": "2015-05-14T17:55:35.000Z",
  "title": "A Variant of the Bombieri-Vinogradov Theorem for Short Intervals With Applications",
  "authors": [
    "Jesse Thorner"
  ],
  "comment": "15 pages. Referee comments now implemented",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove a Chebotarev analogue of the Bombieri-Vinogradov theorem for short intervals. Using this result in conjunction with recent work of Maynard, we prove that Chebotarev primes (determined by a Galois extension $L/\\mathbb{Q}$) exhibit dense clusters in short intervals. We explore several arithmetic applications related to a question of Serre regarding the Fourier coefficients of cuspidal modular forms. These applications include finding dense clusters of fundamental discriminants $ d $ in short intervals for which the central values of $d$-quadratic twists of modular $L$-functions are non-vanishing, or for which $d$-quadratic twists of elliptic curves over $\\mathbb{Q}$ have rank zero.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-07T04:24:25.000Z",
      "abstract": "We prove a short interval version of the Bombieri-Vinogradov Theorem for Chebotarev sets. Appealing to the recent work of Maynard, we obtain dense clusters of Chebotarev primes in short intervals. As an application, we find dense clusters of nonzero Fourier coefficients of cusp forms in short intervals. This implies that there are dense clusters of fundamental discriminants $\\mathfrak{d}$ in short intervals for which the central values of modular $L$-functions twisted by $\\mathfrak{d}$ are non-vanishing, or for which modular elliptic curves twisted by $\\mathfrak{d}$ have rank zero.",
      "comment": "14 pages. Referee comments now implemented",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-05-14T17:55:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N36",
      "11R42",
      "11F30"
    ],
    "keywords": [
      "bombieri-vinogradov theorem",
      "dense clusters",
      "application",
      "nonzero fourier coefficients",
      "short interval version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.5486T"
    }
  }
}