{"paper":{"title":"The Riemann zeta function at the zeros of real-rooted trigonometric polynomials","arxiv_id":"2610.26705","primary_cat":"math.NT","submitted_at":"2026-10-19T16:40:43+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":0.92,"id":"s0","own":false,"role":"background","text":"Let $P$ be a trigonometric polynomial with only real zeros.","hedge":{"p":0.81,"label":"asserted"},"own_p":0.09,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.92,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"result","text":"We prove that at most $\\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $\\zeta(s)$ on the critical line.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.69,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"result","text":"We also show that \\begin{equation*} \\sup_{\\substack{T\\leq x\\leq2T\\\\ P(x)=0}} \\log \\left|\\zeta\\left(\\frac{1}{2}+ix\\right)\\right| \\geq (1+o(1))\\sqrt{\\frac{\\log T}{6\\log\\log T}}.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.18,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.6,"id":"s3","own":false,"role":"interpretation","text":"\\end{equation*} These results extend the corresponding theorems of Li and Radziwill on the zeros of $\\zeta(s)$ along vertical arithmetic progressions.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.23,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.06,"background":0.2,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.6}},{"i":4,"p":0.68,"id":"s4","own":false,"role":"contribution","text":"The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\\sum_{P(\\gamma)=0} {\\bf \\delta}_\\gamma$ is crystalline.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.29,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.32,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.68,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.99,"text":"We prove that at most $\\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $\\zeta(s)$ on the critical line.","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":4515,"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":0.09,"states_limitation":0.31,"human_participants":0.01,"controlled_comparison":0.03,"randomized_assignment":0.01,"states_quantitative_result":0.99,"compares_with_prior_results":0.94,"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":1.0,"text":"The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\\sum_{P(\\gamma)=0} {\\bf \\delta}_\\gamma$ is crystalline.","location":"abstract s4"}},"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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