Robin's Inequality and Superabundant Counterexamples
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00519
curl -X POST /api/v1/records/hexagon/2610.00519/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00519/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'