{
  "id": "1103.4451",
  "version": "v2",
  "published": "2011-03-23T06:11:14.000Z",
  "updated": "2011-05-23T13:45:36.000Z",
  "title": "On the symmetry of the Liouville function in almost all short intervals",
  "authors": [
    "Giovanni Coppola"
  ],
  "comment": "The paper has been withdrawn by the Author because of a crucial error in Lemma 2, due to the wrong Lemma A",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove a kind of \"almost all symmetry\" result for the Liouville function $\\lambda(n):=(-1)^{\\Omega(n)}$, giving non-trivial bounds for its \"symmetry integral\", say $I_{\\lambda}(N,h)$ : we get $I_{\\lambda}(N,h)\\ll NhL^3+Nh^{21/20}$, with $L:=\\log N$. We also give similar results for other related arithmetic functions, like the M\\\"{o}bius function $\\mu(n)$ ($=\\lambda(n)$ on square-free $n$).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-23T13:45:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N25"
    ],
    "keywords": [
      "liouville function",
      "short intervals",
      "symmetry integral",
      "similar results",
      "related arithmetic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.4451C"
    }
  }
}