{
  "id": "1604.01750",
  "version": "v1",
  "published": "2016-04-06T19:51:22.000Z",
  "updated": "2016-04-06T19:51:22.000Z",
  "title": "An explicit bound for the least prime ideal in the Chebotarev density theorem",
  "authors": [
    "Jesse Thorner",
    "Asif Zaman"
  ],
  "comment": "61 pages. This paper subsumes the contents of arXiv:1510.08086 but adds too much new material to be considered a revised version",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of $\\mathbb{F}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-06T19:51:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chebotarev density theorem",
      "prime ideal",
      "explicit bound",
      "primitive binary quadratic form",
      "explicit log-free zero density estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160401750T"
    }
  }
}