{
  "abstract": "We prove the Hardy-Littlewood asymptotic for prime pairs of every fixed positive even difference $h$ from a restricted uniform two-over-two zeta-ratios formula. The formula may have error $O_\\varepsilon(U^{\\vartheta+\\varepsilon})$ for any fixed $1/2\\le\\vartheta<(39-\\sqrt{177})/32$. The real parts of all four shifts lie in $[3/64,5/64]$; their imaginary parts need only have absolute value at most $U^ν$, for a fixed $0<ν\\le1/4$. For a fixed real smooth weight supported in $(1,1+1/(25h))$, we obtain an error $O_{h,W,\\varepsilon}(x^{-Δ(\\vartheta)+\\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$, where $Δ(\\vartheta)=7/16-\\vartheta+\\vartheta/(3-\\vartheta)>0$. Adding the Riemann hypothesis improves this error to $O_{h,W,\\varepsilon}(x^{\\vartheta-7/8+\\varepsilon})$. The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral. Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference $h$. We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_h$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function. The latter is connected to a logarithmically weighted Möbius approximant in the Nyman-Beurling criterion by finite identities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32672",
  "author_total": 1,
  "authors": [
    {
      "name": "SeungJu Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0e6ra02vr0an8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32672",
  "fulltext_url": null,
  "id": "0e6ra02vr0an8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32672",
  "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": "W7223693438",
  "pdf_url": "https://arxiv.org/pdf/2610.32672",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0e6ra02vr0an8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32672",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32672"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Hardy--Littlewood asymptotic for primes $p$ and $p+4$ from a uniform zeta-ratios conjecture",
  "title_text": "The Hardy--Littlewood asymptotic for primes p and p+4 from a uniform zeta-ratios conjecture",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/0e6ra02vr0an8",
  "venue": "arXiv",
  "versions": []
}