{
  "abstract": "For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$. We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $|g|\\in[N,2N]$ have $p_g\\leq(\\log |g|)^{3.44}$ with at most $O((\\log N)^{2.44})$ exceptions. This can be directly compared to the uniform bound $p_g\\leq(\\log |g|)^{19}$ proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other ``almost all\" bounds for $p_g$ that hold unconditionally.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08569",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel R. Johnston"
    },
    {
      "name": "Sunil Naik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d789zxn4nhfp7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08569",
  "fulltext_url": null,
  "id": "d789zxn4nhfp7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08569",
  "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": "W7221059846",
  "pdf_url": "https://arxiv.org/pdf/2610.08569",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d789zxn4nhfp7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08569",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08569"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The least prime with a given primitive root and a variant of the larger sieve",
  "title_text": "The least prime with a given primitive root and a variant of the larger sieve",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/d789zxn4nhfp7",
  "venue": "arXiv",
  "versions": []
}