{"id":"hexagon/2610.00702","pdf":"https://hexagonmath.org/pdf/2610.00702v1","url":"https://hexagonmath.org/abs/2610.00702v1","tags":["math.NA","msc:11M26","msc:11A41","msc:11A25","msc:11N37","Chebyshev function","Colossally abundant numbers","Nicolas' inequality","Riemann Hypothesis","Robin's inequality"],"score":0,"title":"Robin's Inequality at Colossally Abundant Numbers and Nicolas' Inequality at Their Largest Prime Factor","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Frank Vega"],"reviews":0,"subject":"math.NT","summary":"Robin's criterion states that the Riemann Hypothesis holds if and only if $\\sigma(n) 5040$, and Nicolas' criterion states that it holds if and only if $c(q) = \\log(q\\#/\\varphi(q\\#)) - \\gamma - \\log\\log\\theta(q) > 0$ for every prime $q$, where $q\\#$ is the primorial of $q$ and $\\theta$ is the Chebyshev function. Let $N$ be a colossally abundant number, let $\\xi(N)$ be the real number associated with $N$ by Nicolas, and let $q$ be the largest prime not exceeding $\\xi(N)$. Writing $\\Delta(N) = \\log(\\sigma(N)/N) - \\gamma - \\log\\log\\log N$, we prove without any hypothesis that, whenever $\\log q…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}