{
  "abstract": "In 2001 Baker, Harman, and Pintz proved, for $x$ sufficiently large, the existence of at least one prime in the sub-linear interval $[x,x+x^{\\frac{1}{2} +\\frac{1}{40}}]$. But just how large is sufficiently large? In the current note I shall make the Baker--Harman--Pintz theorem effective, but at an extremely high price. Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for $x \\geq 3.5\\times 10^{90}$. Furthermore, again assuming RH, I shall also show that for any $0< m\\leq15.831$ and $x \\geq 1$ there is at least one prime in the interval $[x,x+x^{\\frac{1}{2} +\\frac{1}{m}}]$, and that for $M\\geq 2.29$ and $y \\geq 1$ there is at least one prime in the interval $[y^M, (y+1)^M]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27830",
  "author_total": 1,
  "authors": [
    {
      "name": "Matt Visser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa1j8s1mm36gj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27830",
  "fulltext_url": null,
  "id": "fa1j8s1mm36gj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27830",
  "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": "W7223110437",
  "pdf_url": "https://arxiv.org/pdf/2610.27830",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa1j8s1mm36gj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27830",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27830"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Primes in short intervals assuming the Riemann hypothesis",
  "title_text": "Primes in short intervals assuming the Riemann hypothesis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/fa1j8s1mm36gj",
  "venue": "arXiv",
  "versions": []
}