{
  "id": "1306.5322",
  "version": "v4",
  "published": "2013-06-22T13:27:17.000Z",
  "updated": "2015-11-09T21:06:15.000Z",
  "title": "Explicit formulae for primes in arithmetic progressions, I",
  "authors": [
    "Tomohiro Yamada"
  ],
  "comment": "19 pages, appending an explicit form of BV theorem",
  "categories": [
    "math.NT"
  ],
  "abstract": "We shall give an explicit formula for $\\psi(x, q, a)$ with an error term of the form $C/\\log^\\alpha x$ under the condition that $q<\\log^{\\alpha_1} x$ is nonexceptional, for various values of $\\alpha$ and $\\alpha_1$. We shall also give an explicit formula for $\\psi(x, q, a)$ with error terms $C/\\log^A x$ working whether $q$ is exceptional or nonexceptional, but under the condition that $\\frac{0.4923A}{\\pi}q^{1/2}\\log^2 q<\\log x/\\log\\log x$. Moreover, we shall give an explicit form of Bombieri-Vinogradov theorem over non-exceptional moduli.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-15T03:53:18.000Z",
      "abstract": "We shall give an explicit formula for $\\psi(x, q, l)$ with error terms $\\log\\log x/\\log^\\alpha x$ under the condition that $q<\\log^{\\alpha_1} x$ is nonexceptional, for various values of $\\alpha$ and $\\alpha_1$. We shall also give an explicit formula for $\\psi(x, q, l)$ with error terms $\\log\\log x/\\log^A x$ working whether $q$ is exceptional or nonexceptional, but under the condition that $\\frac{0.4923A}{\\pi}q^{1/2}\\log^2 q<\\log x/\\log\\log x$.",
      "comment": "19 pages, revised some errors and extended main theorems",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-11-09T21:06:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N13"
    ],
    "keywords": [
      "explicit formula",
      "arithmetic progressions",
      "error terms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.5322Y"
    }
  }
}