{"paper":{"title":"Slightly improved zero-free half-planes for the quasi-Riemann hypothesis","arxiv_id":"2610.12234","primary_cat":"math.NT","submitted_at":"2026-10-08T16:18:53+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":"Recently, a uniform $\\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\\mathbb Q(\\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [1].","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.97,"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":0.88,"id":"s1","own":false,"role":"background","text":"In its proof, there are two chosen parameters $b=\\frac{1}{8}, \\ell=\\frac{1}{6}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.09,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.91,"role_probabilities":{"gap":0.0,"method":0.12,"result":0.0,"background":0.88,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.82,"id":"s2","own":true,"role":"result","text":"In this note, via varying $b$ and $\\ell$, following the same strategy, we slightly improve the bound $\\frac{7}{8}=0.875$ to be $$B_{\\mathrm r}=\\frac{34999}{40000}=0.874975, (\\text{when }b_{\\mathrm r}=\\frac{1}{8}, \\ell_{\\mathrm r}=\\frac{1}{6} + \\frac{1}{10000});$$ and \\[ \\begin{gathered} B_{\\mathrm{new}}=\\frac{1507-2\\sqrt{921}}{1653} =0.874957069799\\ldots, (\\text{when }b_{\\mathrm{new}}=-\\frac{4}{29}+\\frac{230\\sqrt{921}}{26709}, \\ell_{\\mathrm{new}}=\\frac{33+8\\sqrt{921}}{1653}).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.93,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.82,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.18,"interpretation":0.0}},{"i":3,"p":0.36,"id":"s3","own":false,"role":"result","text":"\\end{gathered} \\] This result has been formalized by Lean ([15]).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.32,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.17,"result":0.36,"background":0.17,"limitation":0.01,"future_work":0.01,"contribution":0.21,"interpretation":0.07}}]},"compiled":{"main_claim":{"p":1.0,"text":"In this note, via varying $b$ and $\\ell$, following the same strategy, we slightly improve the bound $\\frac{7}{8}=0.875$ to be $$B_{\\mathrm r}=\\frac{34999}{40000}=0.874975, (\\text{when }b_{\\mathrm r}=\\frac{1}{8}, \\ell_{\\mathrm r}=\\frac{1}{6} + \\frac{1}{10000});$$ and \\[ \\begin{gathered} B_{\\mathrm{new}}=\\frac{1507-2\\sqrt{921}}{1653} =0.874957069799\\ldots, (\\text{when }b_{\\mathrm{new}}=-\\frac{4}{29}+\\frac{230\\sqrt{921}}{26709}, \\ell_{\\mathrm{new}}=\\frac{33+8\\sqrt{921}}{1653}).","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":4047,"cost_usd_reported":0.0},"compiled_at":"2026-10-09","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.05,"states_limitation":0.19,"human_participants":0.01,"controlled_comparison":0.13,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.95,"mentions_code_or_data_availability":0.28},"evidential_basis":{"p":1.0,"value":"proof 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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":4,"sentences_in_abstract":4},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}