{
  "abstract": "In this paper we study the least positive integer $G(n)$ such that the integers $a\\leq G(n)$ with $(a,n)=1$ generate $(\\mathbb Z/n\\mathbb Z)^\\times$. We prove that, for every $\\varepsilon>0$, \\[ \\sum_{n\\leq x}G(n)\\ll_\\varepsilon x(\\log x)^{8/3+\\varepsilon}, \\] improving the previously known bound $\\ll x(\\log x)^{97}$. We also prove that $G(n)\\leq(\\log n)^2$ for almost all $n$, unconditionally. Thus, for almost all $n$, we obtain the same logarithmic exponent $2$ as in the classical pointwise bound under GRH.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10177",
  "author_total": 1,
  "authors": [
    {
      "name": "Chiara Bellotti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/44kxwea0sdeh7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10177",
  "fulltext_url": null,
  "id": "44kxwea0sdeh7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10177",
  "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"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221024303",
  "pdf_url": "https://arxiv.org/pdf/2610.10177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/44kxwea0sdeh7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Improved average and almost-all bounds for $G(n)$",
  "title_text": "Improved average and almost-all bounds for G(n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/44kxwea0sdeh7",
  "venue": "arXiv",
  "versions": []
}