{
  "id": "1709.06445",
  "version": "v1",
  "published": "2017-09-18T07:10:08.000Z",
  "updated": "2017-09-18T07:10:08.000Z",
  "title": "An elementary property of correlations",
  "authors": [
    "Giovanni Coppola"
  ],
  "comment": "Assuming Delange Hypothesis(DH), we prove the \"Ramanujan exact explicit formula\" for $f,g$ correlation; for 2k-twin primes, assuming $(DH)$ we prove Hardy-Littlewood Conjecture",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the shift-Ramanujan expansion (see 1705.07193) of general $f,g$ satisfying Ramanujan Conjecture, in order to get formulae, for their shifted convolution sum, say $C_{f,g}(N,a)$, of length $N$ and shift $a$ (so, the Ramanujan expansion is with respect to a>0). We prove that, assuming Delange Hypothesis (DH) for the expansion, we get say Ramnujan exact explicit formula (R.e.e.f.). A noteworthy case, of course, is $f=g=\\Lambda$, the von Mangoldt function, so $C_{\\Lambda,\\Lambda}(N,2k)$, for natural $k$, regards $2k-$twin primes; assuming $(DH)$ for them, we get (from corresponding R.e.e.f.) the proof, easily, of Hardy-Littlewood Conjecture for them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-18T07:10:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11P32",
      "11N37"
    ],
    "keywords": [
      "elementary property",
      "correlations",
      "ramnujan exact explicit formula",
      "von mangoldt function",
      "satisfying ramanujan conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}