{
  "id": "1007.1018",
  "version": "v1",
  "published": "2010-07-07T00:08:50.000Z",
  "updated": "2010-07-07T00:08:50.000Z",
  "title": "On the Symmetry Integral",
  "authors": [
    "Giovanni Coppola"
  ],
  "comment": "Plain TeX(5 pages)",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give a level one result for the \"symmetry integral\", say $I_f(N,h)$, of essentially bounded $f:\\N \\to \\R$; i.e., we get a kind of \"square-root cancellation\" \\thinspace bound for the mean-square (in $N<x\\le 2N$) of the \"symmetry\" \\thinspace of, say, the arithmetic function $f:=g\\ast \\1$, where $g:\\N \\to \\R$ is such that $\\forall \\epsilon>0$ we have $g(n)\\ll_{\\epsilon} n^{\\epsilon}$, and supported in $[1,Q]$, with $Q\\ll N$ (so, the exponent of $Q$ relative to $N$, say the level $\\lambda:=(\\log Q)/(\\log N)$ is $\\lambda < 1$), where the symmetry sum weights the $f-$values in (almost all, i.e. all but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$ (with positive/negative sign at the right/left of $x$), with mild restrictions on $h$ (say, $h\\to \\infty$ and $h=o(\\sqrt N)$, as $N\\to \\infty$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-07T00:08:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N25"
    ],
    "keywords": [
      "symmetry integral",
      "symmetry sum weights",
      "mild restrictions",
      "short intervals",
      "square-root cancellation"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.1018C"
    }
  }
}