{"id":"hexagon/2610.00519","pdf":"https://hexagonmath.org/pdf/2610.00519v1","url":"https://hexagonmath.org/abs/2610.00519v1","tags":["math.NA","msc:11M26","msc:11A41","msc:11A25","Chebyshev function","Prime Numbers","Riemann Hypothesis","Robin's inequality","Squarefree kernel","Sum-of-divisors function"],"score":0,"title":"Robin's Inequality and Superabundant Counterexamples","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 $\\sigma(n) 5040$. Let $n'$ be the squarefree kernel of $n$, $p_{k}$ its largest prime factor, and $d = e^{\\gamma}/\\zeta(2) \\approx 1.08276$. We prove that Robin's inequality holds for every $n > 5040$ with $d \\cdot \\log\\log n' \\leq \\log\\log n$, and for every $n > 5040$ whose prime factors are all below $(\\log n)^{1/d}$, and that both results remain valid with $d$ replaced by $1.016$. Our main results concern superabundant numbers, among which there are infinitely many counterexamples to Robin's inequality if the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}