{
  "id": "1801.08475",
  "version": "v1",
  "published": "2018-01-25T16:36:09.000Z",
  "updated": "2018-01-25T16:36:09.000Z",
  "title": "Explicit formula for the average of Goldbach and prime tuples representations",
  "authors": [
    "Marco Cantarini"
  ],
  "comment": "Submitted",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Lambda\\left(n\\right)$ be the Von Mangoldt function, let \\[ r_{G}\\left(n\\right)=\\underset{{\\scriptstyle m_{1}+m_{2}=n}}{\\sum_{m_{1},m_{2}\\leq n}}\\Lambda\\left(m_{1}\\right)\\Lambda\\left(m_{2}\\right), \\] \\[ r_{PT}\\left(N,h\\right)=\\sum_{n=0}^{N}\\Lambda\\left(n\\right)\\Lambda\\left(n+h\\right),\\,h\\in\\mathbb{N} \\] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\\left(n\\right)$ and $r_{PT}\\left(N,h\\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\\left(a,b\\right)$, series over $\\rho$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of $r_{G}\\left(n\\right)$. Some observation about these formulae and the average with Ces\\`aro weight \\[ \\frac{1}{\\Gamma\\left(k+1\\right)}\\sum_{n\\leq N}r_{G}\\left(n\\right)\\left(N-n\\right)^{k},\\,k>0 \\] are included.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-25T16:36:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "explicit formula",
      "prime tuples representations",
      "counting function",
      "von mangoldt function",
      "incomplete beta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}