{
  "abstract": "Let \\(E(X)\\) count the even integers between \\(4\\) and \\(X\\) that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove \\(E(X)\\ll_\\varepsilon X^{7/12+\\varepsilon}\\) whenever all Dirichlet \\(L\\)-functions have a common fixed zero-free half-plane \\(\\Rea s>θ_0\\), with \\(θ_0<1\\). The uniform \\(7/8\\) theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most \\(θ\\), where \\(1/2\\leqθ\\leq2/3\\), we prove \\[ E(X)\\ll_{θ,\\varepsilon}X^{g(θ)+\\varepsilon}, \\qquad g(θ)=\\frac{θ(5-4θ)}{2(2-θ)}. \\] Thus every fixed \\(θ<2/3\\) gives an exponent strictly below \\(7/12\\). The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15559",
  "author_total": 2,
  "authors": [
    {
      "name": "János Pintz"
    },
    {
      "name": "Ingo Althöfer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2083cp7j9mxc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15559",
  "fulltext_url": null,
  "id": "c2083cp7j9mxc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15559",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221814246",
  "pdf_url": "https://arxiv.org/pdf/2610.15559",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2083cp7j9mxc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15559",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15559"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Goldbach exceptional set: an exponent 7/12 and conditional refinement",
  "title_text": "The Goldbach exceptional set: an exponent 7/12 and conditional refinement",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/c2083cp7j9mxc",
  "venue": "arXiv",
  "versions": []
}