{
  "id": "1703.08032",
  "version": "v1",
  "published": "2017-03-23T12:18:29.000Z",
  "updated": "2017-03-23T12:18:29.000Z",
  "title": "New estimates for some prime functions",
  "authors": [
    "Christian Axler"
  ],
  "comment": "14 pages, comments are welcome. arXiv admin note: text overlap with arXiv:1703.02407",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we first establish new explicit estimates for Chebyshev's $\\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime counting function $\\pi(x)$, which improve the currently best ones. Furthermore, we use the obtained estimates for the prime counting function to give two new results concerning the existence of prime numbers in short intervals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-23T12:18:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11A41"
    ],
    "keywords": [
      "prime functions",
      "prime counting function",
      "prime numbers",
      "short intervals",
      "explicit estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}