{
  "abstract": "Fix $d\\ge1$ and choose a polynomial $P$ uniformly at random from the set of integer polynomials of degree $d$ whose coefficients have absolute value at most $H$. We study $E_{P}$, the error in the count of squarefree values $P(n)$ for $n\\leq x$. In the range $x=H^{o\\left(1\\right)}$, we prove that $E_{P}\\left(x\\right)/x^{\\frac{1}{4}}$ converges to a Gaussian mixture that is not itself Gaussian, $\\sqrt{C\\left(G\\right)}Z$, where $G$ is the weak limit, as $H\\rightarrow\\infty$, of the prime-indexed vector $\\left(P\\bmod p^{2}\\right)_{p}$, $C$ is a nonnegative function and $Z$ is an independent standard Gaussian. This answers a question of Sofos. Finally, in the appendix, we establish that $\\log \\mathbb{E}[C(G)^m]\\asymp_d m^2/(\\log m)^2$ as $m\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29911",
  "author_total": 1,
  "authors": [
    {
      "name": "Omer Simhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd83ea55bpgs1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29911",
  "fulltext_url": null,
  "id": "bd83ea55bpgs1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29911",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223229185",
  "pdf_url": "https://arxiv.org/pdf/2610.29911",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd83ea55bpgs1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29911",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29911"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Squarefree values of random polynomials are Gaussian mixtures",
  "title_text": "Squarefree values of random polynomials are Gaussian mixtures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/bd83ea55bpgs1",
  "venue": "arXiv",
  "versions": []
}