{
  "abstract": "Let $E_\\square(N)$ count the positive integers at most $N$ that cannot be expressed as a prime plus a nonnegative integer square. Assuming that every nontrivial zero of every primitive Dirichlet $L$-function has real part at most $7/8$, we prove $E_\\square(N)\\ll_\\eps N^{11/20+\\eps}$ for every $\\eps>0$. The proof combines a conductor-weighted mean-square estimate with a uniform estimate for certain families having a prescribed quadratic component. For these families, the large sieve depends on the varying part of the conductor, while an approximate functional equation controls the fourth moment using the full conductor. A zero-detection argument then absorbs all bands of zeros with real part at least $129/167$. The remaining bands and the minor arcs give the exponent $11/20$. The proof includes even denominators, completion and positivity of the singular series, and removal of proper prime powers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21655",
  "author_total": 2,
  "authors": [
    {
      "name": "János Pintz"
    },
    {
      "name": "Ingo Althöfer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ck3rs5frgpk5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21655",
  "fulltext_url": null,
  "id": "1ck3rs5frgpk5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21655",
  "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": "W7222762089",
  "pdf_url": "https://arxiv.org/pdf/2610.21655",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ck3rs5frgpk5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21655",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21655"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A prime plus a square: A conditional 11/20 upper bound",
  "title_text": "A prime plus a square: A conditional 11/20 upper bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/1ck3rs5frgpk5",
  "venue": "arXiv",
  "versions": []
}