{
  "abstract": "Fix $r\\ge2$ and let $D_k$ be the natural density of the $n$ such that none of $n+1,\\dots,n+k$ is $r$-free. We prove that $\\log(1/D_k)=\\frac{r-1}{ζ(r)}k\\log k+\\frac{r}{ζ(r)}k\\log\\log k-C_rk+o(k)$, where $C_r=\\frac{1}{ζ(r)}((r-1)\\logζ(r)+r-1+r\\log\\fracπ{r\\sin(π/r)})$; for $r=2$ this is $C_2=\\frac{6}{π^2}(1+\\log\\frac{π^4}{24})=1.4595513\\dots$. This determines Mirsky's constant $α(h)$, the density of the $n$ with $s_{n+1}-s_n=h$ in the squarefree numbers, to the same precision. Previously the only known estimate was $\\logα(h)\\le-\\frac54 h\\log\\log h+O(h)$; ours improves this by a factor of order $\\log h/\\log\\log h$, and is two-sided. The upper bound rests on the observation that the primes $p\\le z$ with $\\prod_{p\\le z}p^r\\le\\sqrt k$ sieve a window of length $k$ with no freedom of alignment, so that at least $(1/ζ(r)+o(1))k$ positions must be covered by $r$-th powers of distinct primes exceeding $k^{1/r}$. The resulting sum over injections is $m!e_m$ with $m\\sim k/ζ(r)$, evaluated by a saddle point. For the lower bound the Chung-Erdos inequality recovers the full contribution of $m!e_m$ rather than merely its largest term, which is what makes the linear coefficient explicit. As a consequence the first-moment threshold for gaps between $r$-free numbers is $\\frac{ζ(r)}{r-1}\\frac{\\log x}{\\log\\log x}(1-\\frac{(1+o(1))\\log\\log\\log x}{(r-1)\\log\\log x})$; for $r=2$ the constant is $π^2/6$, that of Erdos's 1951 lower bound. We also prove that every covering system of length $k$ has modulus at least $\\exp((\\frac{r}{ζ(r)}+o(1))k\\log k)$, so the Chinese remainder construction yields only $ζ(r)/r$, half of $π^2/6$ when $r=2$. Erdos's bound is thus equivalent to the assertion that the first run of $k$ consecutive non-$r$-free integers occurs at the first-moment threshold, not at the scale set by the minimal modulus of a covering system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09221",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyang Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3z9pnht21fn69.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09221",
  "fulltext_url": null,
  "id": "3z9pnht21fn69",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09221",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212140885",
  "pdf_url": "https://arxiv.org/pdf/2609.09221",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3z9pnht21fn69.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.09221",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09221"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The density of long runs of non-$r$-free integers",
  "title_text": "The density of long runs of non-r-free integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/3z9pnht21fn69",
  "venue": "arXiv",
  "versions": []
}