{"paper":{"title":"A finite arithmetic form of Robin's inequality and its equivalence to the Riemann hypothesis","arxiv_id":"2609.26787","primary_cat":"math.NT","submitted_at":"2026-09-22T17:58:00+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":1.0,"id":"s0","own":false,"role":"background","text":"Robin's inequality reformulates the Riemann hypothesis as a bound on the sum-of-divisors function $\\sigma$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.06,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.98,"id":"s1","own":true,"role":"contribution","text":"We introduce a finite-truncation version of Robin's inequality determined intrinsically by the arithmetic structure of $n$, \\[ \\frac{\\sigma(n)}{n} < e^\\gamma \\sum_{j=0}^{\\omega(n)} \\frac{(\\ln\\ln\\ln n)^j}{j!}, \\qquad n>5040, \\] where $\\omega(n)$ denotes the number of distinct prime factors of $n$, and $\\gamma$ is the Euler-Mascheroni constant.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.12,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}},{"i":2,"p":0.98,"id":"s2","own":true,"role":"result","text":"The resulting bound is pointwise stronger than Robin's inequality and we prove it unconditionally for integers with $\\omega(n)\\le 6$, primorials, odd integers and square-free integers.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.14,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":3,"p":0.92,"id":"s3","own":true,"role":"result","text":"We also prove that this inequality is equivalent to Robin's inequality, and hence to the Riemann hypothesis.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.56,"quantitative":false,"quantitative_p":0.23,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.92,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.02}},{"i":4,"p":0.61,"id":"s4","own":true,"role":"interpretation","text":"Consequently, the Riemann hypothesis is equivalent to the truncated inequality holding for all colossally abundant numbers.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.37,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.38,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.61}},{"i":5,"p":0.95,"id":"s5","own":true,"role":"result","text":"We further show that if the inequality fails, its minimal counterexample must be a superabundant number, placing any obstruction within a rigid extremal class of integers.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.95,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.03}}]},"compiled":{"main_claim":{"p":0.84,"text":"We also prove that this inequality is equivalent to Robin's inequality, and hence to the Riemann hypothesis.","location":"abstract s3","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 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determined intrinsically by the arithmetic structure of $n$, \\[ \\frac{\\sigma(n)}{n} < e^\\gamma \\sum_{j=0}^{\\omega(n)} \\frac{(\\ln\\ln\\ln n)^j}{j!}, \\qquad n>5040, \\] where $\\omega(n)$ denotes the number of distinct prime factors of $n$, and $\\gamma$ is the Euler-Mascheroni constant.","location":"abstract s1"}},"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":6,"sentences_in_abstract":6},"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 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