{"paper":{"title":"The least prime with a given primitive root and a variant of the larger sieve","arxiv_id":"2610.08569","primary_cat":"math.NT","submitted_at":"2026-10-06T15:46:27+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":0.71,"id":"s0","own":true,"role":"background","text":"For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$.","hedge":{"p":0.58,"label":"asserted"},"own_p":0.61,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.71,"limitation":0.0,"future_work":0.0,"contribution":0.29,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"result","text":"We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $|g|\\in[N,2N]$ have $p_g\\leq(\\log |g|)^{3.44}$ with at most $O((\\log N)^{2.44})$ exceptions.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.9,"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":0.72,"id":"s2","own":false,"role":"interpretation","text":"This can be directly compared to the uniform bound $p_g\\leq(\\log |g|)^{19}$ proven recently by Fan and Pollack.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.09,"main_claim_p":0.08,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.08,"background":0.19,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.72}},{"i":3,"p":0.7,"id":"s3","own":true,"role":"method","text":"To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest.","hedge":{"p":0.58,"label":"asserted"},"own_p":0.95,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.7,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.3,"interpretation":0.0}},{"i":4,"p":0.69,"id":"s4","own":true,"role":"contribution","text":"In addition to our conditional result, we discuss other ``almost all\" bounds for $p_g$ that hold unconditionally.","hedge":{"p":0.68,"label":"asserted"},"own_p":0.73,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.29,"background":0.0,"limitation":0.01,"future_work":0.01,"contribution":0.69,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $|g|\\in[N,2N]$ have $p_g\\leq(\\log |g|)^{3.44}$ with at most $O((\\log N)^{2.44})$ exceptions.","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":4389,"cost_usd_reported":0.0},"compiled_at":"2026-10-07","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.05,"states_limitation":0.91,"human_participants":0.01,"controlled_comparison":0.13,"randomized_assignment":0.01,"states_quantitative_result":0.99,"compares_with_prior_results":0.98,"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":0.99,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.01,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.99,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.85,"text":"To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest.","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":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; 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}}