{
  "abstract": "Let $λ(n)=(-1)^{Ω(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\\mathcal E_x\\subseteq[1,x]$ with $|\\mathcal E_x\\cap[1,H]|\\ll_A H(\\log x)^{-A}$ $(1\\le H\\le x)$ for every fixed $A>0$, while $\\max_{\\substack{1\\le h\\le x\\ h\\notin\\mathcal E_x}}\\sup_{1\\le y\\le x}\\left|\\sum_{n\\le y}\\frac{λ(n)λ(n+h)}{n}\\right|\\ll(\\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\\in\\mathbb N$ and $1\\le y\\le x$, $\\left|\\sum_{n\\le y}\\frac{λ(n)λ(n+h)}{n}\\right|\\le\\log(2\\min{h,y})+O((\\log x)^{1-c_{\\mathrm G}})$ for an absolute $c_{\\mathrm G}>0$, with no exceptional shifts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23500",
  "author_total": 1,
  "authors": [
    {
      "name": "Jizhou Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ayeyrg3ebdks7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23500",
  "fulltext_url": null,
  "id": "ayeyrg3ebdks7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23500",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204177186",
  "pdf_url": "https://arxiv.org/pdf/2608.23500",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ayeyrg3ebdks7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.23500",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23500"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Logarithmic Chowla Correlations Across All Shift Scales",
  "title_text": "Logarithmic Chowla Correlations Across All Shift Scales",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/ayeyrg3ebdks7",
  "venue": "arXiv",
  "versions": []
}