{
  "id": "2108.10878",
  "version": "v1",
  "published": "2021-08-24T17:59:01.000Z",
  "updated": "2021-08-24T17:59:01.000Z",
  "title": "Refinements to the prime number theorem for arithmetic progressions",
  "authors": [
    "Jesse Thorner",
    "Asif Zaman"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-\\delta}$, a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region and a refinement of Bombieri's \"repulsive\" log-free zero density estimate. Improvements exist when the modulus is sufficiently powerful.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-24T17:59:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N13",
      "11M06"
    ],
    "keywords": [
      "prime number theorem",
      "arithmetic progression",
      "refinement",
      "log-free zero density estimate",
      "vinogradov-korobov zero-free region"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}