{"paper":{"title":"Large zeta sums and zeros of the Riemann zeta function","arxiv_id":"2608.31060","primary_cat":"math.NT","submitted_at":"2026-08-31T16:38:47+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":"For real $t$ and $x\\ge 1$, set \\[ S(x,t)=\\sum_{n\\le x} n^{\\ii t}.","hedge":{"p":0.59,"label":"unclear"},"own_p":0.36,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.49,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.0,"background":0.92,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":1,"p":0.66,"id":"s1","own":true,"role":"contribution","text":"\\] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.87,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.34,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.66,"interpretation":0.0}},{"i":2,"p":0.99,"id":"s2","own":true,"role":"result","text":"More precisely, if $T\\le |t|\\le 2T$, $\\exp(\\sqrt{\\log T})\\le x\\le \\sqrt T$, and $|S(x,t)|=x/N$ with $N\\le (\\log x)^{1/100}$, then for every $cN^6\\le L\\le (\\log x)/2$ a disk centered at $1+\\ii\\phi$, where $|\\phi-t|\\ll N$, contains at least $L/360$ zeros of $\\zeta(s)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.63,"quantitative":true,"quantitative_p":0.99,"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":3,"p":0.6,"id":"s3","own":true,"role":"interpretation","text":"As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line $\\Ree s=1$ yields $S(x,t)\\ll x/(\\log x)^{1/100}$ in polynomial ranges of $x$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.86,"main_claim_p":0.23,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.39,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.6}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"method","text":"The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.64,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":0.96,"id":"s5","own":true,"role":"method","text":"In the zeta setting the spectral height is shifted by $t$, and an additional residue from the pole of $\\zeta(s)$ appears in the Gaussian transform; in the range considered here that residue is exponentially small.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.72,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.82,"role_probabilities":{"gap":0.0,"method":0.96,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.01}}]},"compiled":{"main_claim":{"p":0.98,"text":"\\] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$.","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":5342,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.04,"states_limitation":0.33,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.63,"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":"\\] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$.","location":"abstract s1"}},"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}}