{
  "id": "1412.0574",
  "version": "v1",
  "published": "2014-12-01T18:29:52.000Z",
  "updated": "2014-12-01T18:29:52.000Z",
  "title": "Gaps of Smallest Possible Order between Primes in an Arithmetic Progression",
  "authors": [
    "Roger C. Baker",
    "Liangyi Zhao"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $t \\in \\mathbb{N}$, $\\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \\leq x^{5/12-\\eta}$, $q$ not a multiple of the conductor of the exceptional character $\\chi^*$ (if it exists). Suppose further that, \\[ \\max \\{p : p | q \\} < \\exp (\\frac{\\log x}{C \\log \\log x}) \\; \\; {and} \\; \\; \\prod_{p | q} p < x^{\\delta}, \\] where $C$ and $\\delta$ are suitable positive constants depending on $t$ and $\\eta$. Let $a \\in \\mathbb{Z}$, $(a,q)=1$ and \\[ \\mathcal{A} = \\{n \\in (x/2, x]: n \\equiv a \\pmod{q} \\} . \\] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $\\mathcal{A}$ with \\[ p_t - p_1 \\ll qt \\exp (\\frac{40 t}{9-20 \\theta}) . \\] Here $\\theta = (\\log q) / \\log x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-01T18:29:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N13"
    ],
    "keywords": [
      "arithmetic progression",
      "sufficiently large real number",
      "natural number",
      "exceptional character",
      "suitable positive constants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0574B"
    }
  }
}