{
  "id": "2303.08277",
  "version": "v1",
  "published": "2023-03-14T23:52:28.000Z",
  "updated": "2023-03-14T23:52:28.000Z",
  "title": "On Erdős sums of almost primes",
  "authors": [
    "Ofir Gorodetsky",
    "Jared Duker Lichtman",
    "Mo Dick Wong"
  ],
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "In 1935, Erd\\H{o}s proved the sums $f_k=\\sum_n 1/(n\\log n)$, over integers $n$ with exactly $k$ prime factors, are uniformly bounded, and in 1993 Zhang proved $f_k$ is maximized by the prime sum $f_1=\\sum_p 1/(p\\log p)$. According to a 2013 conjecture of Banks and Martin, the sums $f_k$ are predicted to decrease monotonically in $k$. In this article, we show the sums restricted to odd integers are indeed monotonically decreasing in $k$, sufficiently large. By contrast, contrary to the conjecture we prove the sums $f_k$ increase monotonically in $k$, sufficiently large. Our main result gives an asymptotic for $f_k$ which identifies the (negative) secondary term, i.e. $f_k = 1 - (a+o(1))k^2/2^k$ for an explicit constant $a= .0656\\cdots$. This is proven by a refined method combining real and complex analysis, while the classical results of Sathe and Selberg on products of $k$ primes imply the weaker estimate $f_k=1+O_{\\varepsilon}(k^{\\varepsilon-1/2})$. We also give an alternate, probability-theoretic argument related to the Dickman distribution. Here the proof reduces to showing a new sequence of integrals converges to $e^{-\\gamma}$, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-14T23:52:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11Y60",
      "11A05",
      "60G18",
      "60H25"
    ],
    "keywords": [
      "erdős sums",
      "sufficiently large",
      "independent interest",
      "conjecture",
      "odd integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}