{
  "abstract": "Let $λ$ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k $. that is, for every fixed even integer $k\\geq 2$, we prove that \\[ \\lim_{\\min\\{M,N\\}\\to \\infty}\\frac{1}{MN}\\sum_{m=1}^M\\sum_{n=1}^Nλ(m)λ(m+a_1n)\\cdotsλ(m+a_kn)=0. \\] We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12869",
  "author_total": 3,
  "authors": [
    {
      "name": "Qiang Huo"
    },
    {
      "name": "Zhengxing Lian"
    },
    {
      "name": "Kangbo Ouyang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/747th20v30jgk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12869",
  "fulltext_url": null,
  "id": "747th20v30jgk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12869",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221553082",
  "pdf_url": "https://arxiv.org/pdf/2610.12869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/747th20v30jgk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Odd order cases of Chowla's averages over arbitrary rectangles",
  "title_text": "Odd order cases of Chowla's averages over arbitrary rectangles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/747th20v30jgk",
  "venue": "arXiv",
  "versions": []
}