{"paper":{"title":"On a series related to the M\\\"obius function","arxiv_id":"2610.23660","primary_cat":"math.NT","submitted_at":"2026-10-16T15:55:06+00:00","version_compiled":"v1"},"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 v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor.","hedge":{"p":0.86,"label":"asserted"},"own_p":0.12,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.29,"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":1.0,"id":"s1","own":false,"role":"background","text":"Alladi and Johnson proved that $\\sum_{P^{-}(n)\\equiv l \\Mod{k}} \\frac{\\mu(n)\\omega(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\\varepsilon\\bigl((\\log\\log X)^{5/3+\\varepsilon}/\\log X\\bigr)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.02,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.95,"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":2,"p":0.95,"id":"s2","own":true,"role":"result","text":"In this article, we prove that these partial sums are in fact equal to $C(k,l)/\\log X + O\\bigl((\\log\\log X)^4/(\\log X)^2\\bigr)$, with an explicit constant $C(k,l)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.91,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.95,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"contribution","text":"We also investigate the behaviour of these constants.","hedge":{"p":0.62,"label":"asserted"},"own_p":0.9,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"result","text":"In particular, we show under GRH that the number of reduced classes $l\\Mod{k}$ for which $C(k,l)<0$ is $\\varphi(k)(1- o(1))$ as $k\\to\\infty$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.95,"main_claim_p":0.13,"quantitative":true,"quantitative_p":0.96,"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}}]},"compiled":{"main_claim":{"p":1.0,"text":"In this article, we prove that these partial sums are in fact equal to $C(k,l)/\\log X + O\\bigl((\\log\\log X)^4/(\\log X)^2\\bigr)$, with an explicit constant $C(k,l)$.","location":"abstract s2","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":4496,"cost_usd_reported":0.0},"compiled_at":"2026-10-20","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.06,"states_limitation":0.65,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.95,"mentions_code_or_data_availability":0.02},"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":1.0,"text":"In this article, we prove that these partial sums are in fact equal to $C(k,l)/\\log X + O\\bigl((\\log\\log X)^4/(\\log X)^2\\bigr)$, with an explicit constant $C(k,l)$.","location":"abstract s2"}},"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 v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":5,"sentences_in_abstract":5},"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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