{"paper":{"title":"An asymptotic Jensen-Polya criterion for GRH","arxiv_id":"2610.38870","primary_cat":"math.NT","submitted_at":"2026-10-28T20:40:35+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 Jensen--P\\'olya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.06,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.05,"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":0.6,"id":"s1","own":false,"role":"background","text":"Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.44,"role_probabilities":{"gap":0.4,"method":0.0,"result":0.0,"background":0.6,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.97,"id":"s2","own":true,"role":"contribution","text":"Here we study the unshifted family as the degree tends to infinity.","hedge":{"p":0.7,"label":"asserted"},"own_p":0.78,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.01,"method":0.02,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.97,"interpretation":0.0}},{"i":3,"p":0.99,"id":"s3","own":true,"role":"result","text":"For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\\sqrt{d\\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.89,"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":4,"p":0.56,"id":"s4","own":true,"role":"result","text":"Therefore, the proportion of real zeros tends to one with error $O_K(\\sqrt{\\log d/d})$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.26,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.56,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.44}},{"i":5,"p":0.56,"id":"s5","own":false,"role":"method","text":"The main theorems have been autoformalized in Lean~4 by AxiomProver.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.47,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.56,"result":0.05,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.38,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.99,"text":"For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\\sqrt{d\\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$.","location":"abstract s3","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":4966,"cost_usd_reported":0.0},"compiled_at":"2026-10-30","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.09,"states_limitation":0.28,"human_participants":0.02,"controlled_comparison":0.04,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.94,"mentions_code_or_data_availability":0.23},"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":0.97,"text":"For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\\sqrt{d\\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$.","location":"abstract s3"}},"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":6,"sentences_in_abstract":6},"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}}