{
  "abstract": "We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with positive integral slopes and nonzero determinant Delta=a_1 b_2-a_2 b_1, put h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2). For either the Liouville or Mobius function in each position and 0<epsilon<=1/10, the correlation sum over x/omega<n<=x, weighted by 1/n, has absolute value at most epsilon log omega whenever x>=omega>=exp_4(C h epsilon^(-2)), where exp_4 denotes four iterated exponentials. A translation lemma separates the common translation from the tower: for lambda(n+b)lambda(n+b+k), with k nonzero, cancellation is uniform when |k|=o(log_4 omega) and |b|=omega^(o(1)), where log_4 is the fourth iterated logarithm. The same thresholds give logarithmic equidistribution of the four Liouville sign pairs. The constant C is effective but depends on uncomputed constants in two analytic inputs. Separately, writing ell=log log x, we prove fully explicit bounds ell/3-(4/3)log ell-7 <= N_lambda(x) <= ell-log ell+2.53/ell for ell>=64, where N_lambda is the non-pretentiousness level in Tao's hypothesis. Under GRH the lower bound improves to ell-log ell-log 2-2/ell; hence N_lambda(x)=ell-log ell+O(1). The arguments use an explicit zero-free region, Harnack's inequality, a finite resonator, and a comparison of smoothed Euler products. These results concern logarithmic averages, not natural-density two-point Chowla correlations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28526",
  "author_total": 1,
  "authors": [
    {
      "name": "Scott D. Hughes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aw0nvx5gatnhc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28526",
  "fulltext_url": null,
  "id": "aw0nvx5gatnhc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28526",
  "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": "W7214287312",
  "pdf_url": "https://arxiv.org/pdf/2609.28526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aw0nvx5gatnhc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Effective logarithmic two-point Chowla bounds in every window",
  "title_text": "Effective logarithmic two-point Chowla bounds in every window",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/aw0nvx5gatnhc",
  "venue": "arXiv",
  "versions": []
}