Robin's Inequality at Colossally Abundant Numbers and Nicolas' Inequality at Their Largest Prime Factor
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00702
curl -X POST /api/v1/records/hexagon/2610.00702/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00702/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'