{"id":"W2963431454","doi":"10.48550/arxiv.0912.4908","title":"Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula\\n for the densities","year":2009,"lang":"","type":"article","venue":"arXiv (Cornell University)","topic":"Analytic Number Theory Research","field":"Mathematics","cited_by":28,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of British Columbia; Université de Montréal","funders":"","keywords":"Mathematics; Modulo; Prime number theorem; Combinatorics; Logarithm; Dirichlet distribution; Prime power; Asymptotic formula; Prime (order theory); Number theory; Analytic number theory; Prime number; Discrete mathematics; Mathematical analysis","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.008311773,0.0007157499,0.001138446,0.002776736,0.001883028,0.002892898,0.003002545,0.001439003,0.008088866],"category_scores_gemma":[0.05209931,0.0007263456,0.001393293,0.001274528,0.008014734,0.009296292,0.004508007,0.005690908,0.001190905],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.003999315,"about_ca_system_score_gemma":0.001187301,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001958988,"about_ca_topic_score_gemma":0.001248333,"domain_scores_codex":[0.9969788,0.00102324,0.0001017769,0.0006676407,0.0009158846,0.0003128302],"domain_scores_gemma":[0.9823676,0.01248532,0.0008645704,0.002030924,0.001448872,0.0008026996],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00002309677,0.00001173185,0.0005826275,0.00002630038,0.000007344337,0.00004198286,0.0001722072,0.002956799,0.0003291562,0.9887342,0.001108268,0.006006202],"study_design_scores_gemma":[0.00001163176,0.00002077468,0.0006118175,0.00003871883,0.0000110959,0.0001263939,0.00003774732,0.08117623,0.0006678759,0.9142789,0.002993104,0.00002572357],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.2087888,0.003155127,0.6395386,0.006856615,0.0006417689,0.0001194359,0.000368122,0.001205908,0.1393257],"genre_scores_gemma":[0.9042993,0.002051503,0.0645099,0.001524236,0.001165457,0.0005306261,0.000335012,0.0005999916,0.02498408],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.008311773,"threshold_uncertainty_score":0.04395741,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.142578702723784,"score_gpt":0.2743528426147986,"score_spread":0.1317741398910146,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}