{"paper":{"title":"Logarithmic Chowla Correlations Across All Shift Scales","arxiv_id":"2608.23500","primary_cat":"math.NT","submitted_at":"2026-08-24T17:03:54+00:00","version_compiled":"v4"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v4","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"Let $\\lambda(n)=(-1)^{\\Omega(n)}$ be the Liouville function.","hedge":{"p":0.88,"label":"asserted"},"own_p":0.05,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.53,"id":"s1","own":true,"role":"contribution","text":"We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.79,"quantitative":true,"quantitative_p":0.81,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.47,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.53,"interpretation":0.0}},{"i":2,"p":0.99,"id":"s2","own":true,"role":"result","text":"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 $\\max_{\\substack{1\\le h\\le x\\ h\\notin\\mathcal E_x}}\\sup_{1\\le y\\le x}\\left|\\sum_{n\\le y}\\frac{\\lambda(n)\\lambda(n+h)}{n}\\right|\\ll(\\log x)^{1-c}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.76,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":3,"p":0.89,"id":"s3","own":true,"role":"result","text":"The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.89,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.03}},{"i":4,"p":0.48,"id":"s4","own":true,"role":"interpretation","text":"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 $\\log x$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.75,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.74,"role_probabilities":{"gap":0.1,"method":0.0,"result":0.03,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.38,"interpretation":0.48}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"method","text":"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.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.9,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.93,"id":"s6","own":true,"role":"method","text":"Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.79,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.93,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.01}},{"i":7,"p":0.92,"id":"s7","own":true,"role":"result","text":"Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\\in\\mathbb N$ and $1\\le y\\le x$, $\\left|\\sum_{n\\le y}\\frac{\\lambda(n)\\lambda(n+h)}{n}\\right|\\le\\log(2\\min{h,y})+O((\\log x)^{1-c_{\\mathrm G}})$ for an absolute $c_{\\mathrm G}>0$, with no exceptional shifts.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.94,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.92,"background":0.0,"limitation":0.03,"future_work":0.0,"contribution":0.05,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.87,"text":"We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range.","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":6733,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.15,"states_limitation":0.61,"human_participants":0.01,"controlled_comparison":0.03,"randomized_assignment":0.01,"states_quantitative_result":0.97,"compares_with_prior_results":0.96,"mentions_code_or_data_availability":0.01},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.6,"text":"We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range.","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v4"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":8,"sentences_in_abstract":8},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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