{
  "abstract": "We revisit the generalized Rayleigh quotient optimization problem related to the Maynard-Tao sieve with the goal of finding a more efficient function basis. We demonstrate that a family based on products of reciprocals, previously used in asymptotic estimates for $H_m = \\liminf_{n \\to \\infty}(p_{n+m}-p_n)$, provides results that are close to the theoretical limits of the sieve across many regimes. In particular, we establish $H_2 \\leq 31 \\, 140$, $H_3 \\leq 2 \\, 560 \\, 022$, $H_4 \\leq 193 \\, 672 \\, 056$, and $H_5 \\leq 13 \\, 659 \\, 045 \\, 840$. We also show that, as $k \\to \\infty$, the related variational quantity $M_k$, introduced by Maynard, satisfies the lower bound $M_k \\geq \\log k - 0.273572 + o(1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41113",
  "author_total": 1,
  "authors": [
    {
      "name": "Paweł Lewulis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vrhqw6jt2p9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41113",
  "fulltext_url": null,
  "id": "3vrhqw6jt2p9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41113",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41113",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vrhqw6jt2p9c.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41113",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41113"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Clusters of primes revisited",
  "title_text": "Clusters of primes revisited",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/3vrhqw6jt2p9c",
  "venue": "arXiv",
  "versions": []
}