{"id":"W2069255299","doi":"10.1016/j.jnt.2005.05.008","title":"A generating function of higher-dimensional Apostol–Zagier sums and its reciprocity law","year":2006,"lang":"en","type":"article","venue":"Journal of Number Theory","topic":"Advanced Mathematical Identities","field":"Mathematics","cited_by":7,"is_retracted":false,"has_abstract":false,"ca_institutions":"Queen's University","funders":"","keywords":"Mathematics; Reciprocity law; Reciprocity (cultural anthropology); Pure mathematics; Dedekind sum; Trigonometry; Dedekind cut; 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":"codex-gemma-dda1882f352a","candidate_categories":["insufficient_payload"],"consensus_categories":[],"category_scores_codex":[0.0007816968,0.0001293137,0.0003527395,0.00004585746,0.00008175307,0.00002220773,0.00008443788,0.00007964779,0.0009205942],"category_scores_gemma":[0.0003669879,0.0001015809,0.0001099896,0.00005938621,0.0001043955,0.0003352335,0.00005416158,0.0002142465,0.00001149876],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0000409528,"about_ca_system_score_gemma":0.00002602215,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000004386224,"about_ca_topic_score_gemma":0.000007860181,"domain_scores_codex":[0.998606,0.0001301508,0.0006563884,0.00009773028,0.0003618969,0.0001477961],"domain_scores_gemma":[0.9977783,0.001092381,0.0006307484,0.00011231,0.0003299879,0.00005629821],"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":[0.0001279334,0.0001743503,0.00005022231,0.0001856395,0.00007541604,0.00001603979,0.0001234686,0.00005798701,0.01321827,0.9813288,0.00444353,0.0001983452],"study_design_scores_gemma":[0.0004266857,0.00006140249,0.0001401813,0.0002037125,0.00010225,0.000158558,0.00007007596,0.00007975989,0.004733127,0.9929524,0.0009668944,0.0001049752],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9772632,0.000638688,0.005553155,0.00009635628,0.0003588691,0.0001225081,0.00001876365,0.00002629453,0.01592213],"genre_scores_gemma":[0.9502884,0.000008427207,0.03987713,0.0001314548,0.0004799252,0.00000349023,0.000002293209,0.00003782527,0.009171016],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.03432398,"threshold_uncertainty_score":0.9999927,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.0299946409880202,"score_gpt":0.3068938390790898,"score_spread":0.2768991980910696,"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."}}