{
  "id": "1003.4553",
  "version": "v2",
  "published": "2010-03-23T23:06:23.000Z",
  "updated": "2010-06-06T16:36:27.000Z",
  "title": "On some lower bounds of some symmetry integrals",
  "authors": [
    "Giovanni Coppola"
  ],
  "comment": "PlainTeX(10 p.).Improved results,corrected previous(v1)Lemma (on discrete mean-square,not integral!)",
  "journal": "Afr. Mat. 25 issue 1 (2014), 183-195",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the \\lq \\lq symmetry integral\\rq \\rq, \\thinspace say $I_f$, of some arithmetic functions $f:\\N \\rightarrow \\R$; we obtain from lower bounds of $I_f$ (for a large class of arithmetic functions $f$) lower bounds for the \\lq \\lq Selberg integral\\rq \\rq \\thinspace of $f$, say $J_f$ (both these integrals give informations about $f$ in almost all the short intervals $[x-h,x+h]$, when $N\\le x\\le 2N$). In particular, when \\thinspace $f=d_k$, the divisor function (having Dirichlet series \\thinspace $\\zeta^k$, with \\thinspace $\\zeta$ \\thinspace the Riemann zeta function), where $k\\ge 3$ is integer, we give lower bounds for the Selberg integrals, say \\thinspace $J_k=J_{d_k}$, of the \\thinspace $d_k$. We apply elementary methods (Cauchy inequality to get Large Sieve type bounds) in order to give $I_f$ lower bounds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-06-06T16:36:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N25"
    ],
    "keywords": [
      "lower bounds",
      "symmetry integrals",
      "arithmetic functions",
      "large sieve type bounds",
      "riemann zeta function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.4553C"
    }
  }
}