{
  "id": "2009.05000",
  "version": "v2",
  "published": "2020-09-10T17:09:59.000Z",
  "updated": "2020-09-11T15:34:25.000Z",
  "title": "Primes in short intervals: Heuristics and calculations",
  "authors": [
    "Andrew Granville",
    "Allysa Lumley"
  ],
  "comment": "37 pages, 12 figures, added a dedication",
  "categories": [
    "math.NT"
  ],
  "abstract": "We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\\ll (\\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\\log x$ to $(\\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2020-09-11T15:34:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N35",
      "11B83"
    ],
    "keywords": [
      "short intervals",
      "calculations",
      "precise conjectures",
      "maximum grows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}