{
  "id": "1009.6121",
  "version": "v9",
  "published": "2010-09-29T11:06:17.000Z",
  "updated": "2011-05-30T10:14:50.000Z",
  "title": "On the symmetry of primes",
  "authors": [
    "Giovanni Coppola"
  ],
  "comment": "The paper has been withdrawn by the Author see 1103.4451v2 comments",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove a kind of \"almost all symmetry\" result for the primes, i.e. we give non-trivial bounds for the \"symmetry integral\", say $I_{\\Lambda}(N,h)$, of the von Mangoldt function $\\Lambda(n)$ ($:= \\log p$ for prime-powers $n=p^r$, 0 otherwise). We get $I_{\\Lambda}(N,h)\\ll NhL^5+Nh^{21/20}L^2$, with $L:=\\log N$; as a Corollary, we bound non-trivially the Selberg integral of the primes, i.e. the mean-square of $\\sum_{x<n\\le x+h}\\Lambda(n)-h$, over $x\\in [N,2N]$, to get the \"Prime Number Theorem in almost all short intervals\" of (log-powers!) length $h\\ge L^{11/2+\\epsilon}$. We trust here in the improvement of the exponent, say $c<11/2$.",
  "revisions": [
    {
      "version": "v9",
      "updated": "2011-05-30T10:14:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N25"
    ],
    "keywords": [
      "von mangoldt function",
      "prime number theorem",
      "short intervals",
      "non-trivial bounds",
      "selberg integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.6121C"
    }
  }
}