{
  "id": "2109.08757",
  "version": "v1",
  "published": "2021-09-17T20:42:02.000Z",
  "updated": "2021-09-17T20:42:02.000Z",
  "title": "A Dynamical Approach to the Asymptotic Behavior of the Sequence $Ω(n)$",
  "authors": [
    "Kaitlyn Loyd"
  ],
  "comment": "19 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We study the asymptotic behavior of the sequence $\\{\\Omega(n) \\}_{ n \\in \\mathbb{N} }$ from a dynamical point of view, where $\\Omega(n)$ denotes the number of prime factors of $n$ counted with multiplicity. First, we show that for any non-atomic ergodic system $(X, \\mathcal{B}, \\mu, T)$, the operators $T^{\\Omega(n)}: \\mathcal{B} \\to L^1(\\mu)$ have the strong sweeping-out property. In particular, this implies that the Pointwise Ergodic Theorem does not hold along $\\Omega(n)$. Second, we show that the behaviors of $\\Omega(n)$ captured by the Prime Number Theorem and Erd\\H{o}s-Kac Theorem are disjoint, in the sense that their dynamical correlations tend to zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-17T20:42:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "dynamical approach",
      "prime number theorem",
      "strong sweeping-out property",
      "non-atomic ergodic system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}