{"id":"W2111247341","doi":"10.3390/math1010009","title":"ρ — Adic Analogues of Ramanujan Type Formulas for 1/π","year":2013,"lang":"en","type":"article","venue":"Mathematics","topic":"Advanced Mathematical Identities","field":"Mathematics","cited_by":14,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Calgary","funders":"Banff International Research Station for Mathematical Innovation and Discovery; National Science Foundation","keywords":"Ramanujan's sum; Mathematics; Complex multiplication; Pure mathematics; Modular form; Type (biology); Elliptic curve; Multiplication (music); Algebraic number; Congruence (geometry); Modular equation; Algebra over a field; Combinatorics; Mathematical analysis; Geometry","routes":{"ca_aff":true,"ca_fund":true,"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":[],"consensus_categories":[],"category_scores_codex":[0.0003087816,0.0002313283,0.0005965379,0.0001224509,0.00007062081,0.00004426779,0.0003304023,0.0001107836,0.0004639432],"category_scores_gemma":[0.002218071,0.0001898899,0.0001759091,0.0002009974,0.0001370017,0.0002619512,0.00009400003,0.00009482643,0.0001915841],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00003714327,"about_ca_system_score_gemma":0.00002929372,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000006382133,"about_ca_topic_score_gemma":0.00000749371,"domain_scores_codex":[0.9983634,0.00001771041,0.0007657786,0.0001876079,0.0003070588,0.000358432],"domain_scores_gemma":[0.9970197,0.001517151,0.0003714596,0.000582048,0.0004142266,0.00009542079],"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":[0.000007814532,0.0004456126,0.00003039045,0.003608085,0.0001109556,8.872298e-7,0.001832668,0.000005854096,0.003520936,0.9731243,0.01681743,0.0004951156],"study_design_scores_gemma":[0.0003375579,0.0001313341,0.00001594678,0.000180759,0.00009314936,0.000007600641,0.0008748258,0.002978233,0.009903719,0.9848253,0.0004463721,0.0002051712],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"methods","genre_scores_codex":[0.5186023,0.0002605884,0.4427344,0.0003342985,0.0003981556,0.004264125,0.00005839736,0.0005398972,0.03280783],"genre_scores_gemma":[0.2451157,0.0000219166,0.7432874,0.00004585587,0.0001031245,0.0002576494,0.0000141779,0.000113034,0.01104113],"genre_candidate":"methods","genre_consensus":null,"teacher_disagreement_score":0.300553,"threshold_uncertainty_score":0.7743486,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.08843062288045941,"score_gpt":0.3535056060161922,"score_spread":0.2650749831357329,"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."}}