{"id":"W2145644873","doi":"10.1142/s1793042113500292","title":"EULER PRODUCTS IN RAMANUJAN'S LOST NOTEBOOK","year":2013,"lang":"en","type":"article","venue":"International Journal of Number Theory","topic":"Advanced Mathematical Identities","field":"Mathematics","cited_by":3,"is_retracted":false,"has_abstract":true,"ca_institutions":"Carleton University","funders":"National Research Foundation of Korea","keywords":"Ramanujan's sum; Mathematics; Ramanujan tau function; Arithmetic function; Euler's formula; Mathematical proof; Pure mathematics; Modular form; Sketch; Ramanujan theta function; Dirichlet series; Algebra over a field; Arithmetic; Discrete mathematics; Dirichlet distribution; Algorithm; 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":[0.001409363,0.0007310932,0.0004186633,0.001358664,0.001627569,0.002488535,0.0007773538,0.000651186,0.01332313],"category_scores_gemma":[0.004797013,0.0003448195,0.0003980087,0.001124466,0.003703095,0.006506389,0.001848427,0.004434658,0.004143986],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001499588,"about_ca_system_score_gemma":0.0009615357,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0009552048,"about_ca_topic_score_gemma":0.0007298889,"domain_scores_codex":[0.9990784,0.0002481665,0.00004361146,0.0001335474,0.0004012363,0.00009505454],"domain_scores_gemma":[0.9990557,0.0004286701,0.00005583926,0.0001420952,0.0002456916,0.00007205376],"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":[0.000007816185,0.000004515881,0.00002635187,0.00002796549,0.00000167819,0.00003802439,0.0001552152,0.00009254845,0.0001469655,0.9836985,0.009184694,0.006615768],"study_design_scores_gemma":[0.00000496156,0.00001286705,0.0001095245,0.00005104892,0.000004515342,0.0001646183,0.00008040947,0.0004470558,0.0007743742,0.8366066,0.1617303,0.00001357456],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"empirical","genre_scores_codex":[0.03268156,0.0186383,0.1464488,0.01542682,0.01049845,0.00006735859,0.0005940366,0.0007185242,0.7749262],"genre_scores_gemma":[0.535276,0.01668148,0.08076905,0.006852599,0.005343479,0.0001524724,0.0005620811,0.001137184,0.3532256],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.01332313,"threshold_uncertainty_score":0.04457033,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.0314863558170211,"score_gpt":0.3286930005312351,"score_spread":0.297206644714214,"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."}}