{"paper":{"title":"Infinite log-concavity of the Taylor coefficients of the Riemann xi-function","arxiv_id":"2610.04972","primary_cat":"math.CO","submitted_at":"2026-10-04T05:42:15+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":"The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre--P\\'olya class.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.08,"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":0.99,"id":"s1","own":false,"role":"background","text":"Br\\\"and\\'en [J. Reine Angew.","hedge":{"p":0.98,"label":"unclear"},"own_p":0.03,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.0,"background":0.99,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"Math., 2011] proved that if an entire function in the Laguerre--P\\'olya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.11,"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":3,"p":0.67,"id":"s3","own":false,"role":"background","text":"Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\\lambda_n)_{n\\ge0}$.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.11,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.04,"background":0.67,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.27}},{"i":4,"p":0.85,"id":"s4","own":true,"role":"result","text":"In this paper, we prove that the sequence $(\\lambda_n)_{n\\ge0}$ is strictly infinitely log-concave.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.94,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.85,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.15,"interpretation":0.0}},{"i":5,"p":0.48,"id":"s5","own":true,"role":"interpretation","text":"This resolves a conjecture of Zhu [Math. Z., 2023].","hedge":{"p":1.0,"label":"asserted"},"own_p":0.69,"main_claim_p":0.13,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.06,"method":0.0,"result":0.29,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.16,"interpretation":0.48}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"method","text":"The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.92,"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}}]},"compiled":{"main_claim":{"p":1.0,"text":"In this paper, we prove that the sequence $(\\lambda_n)_{n\\ge0}$ is strictly infinitely log-concave.","location":"abstract s4","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":5529,"cost_usd_reported":0.0},"compiled_at":"2026-10-06","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.09,"states_limitation":0.08,"human_participants":0.01,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.11,"compares_with_prior_results":0.8,"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 paper, we prove that the sequence $(\\lambda_n)_{n\\ge0}$ is strictly infinitely log-concave.","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":7,"sentences_in_abstract":7},"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}}