extraordinary number
Define the function $G$ for integers $n>1$ by
$$G(n):=\frac{\sigma (n)}{n\mathrm{ln}(\mathrm{ln}n)},$$ |
where $\sigma (n)$ is the sum of the positive divisors^{} of $n$. A positive integer $N$ is said to be an extraordinary number if it is composite and
$$G(N)\ge \mathrm{max}\{G(N/p),G(aN)\}$$ |
for any prime factor^{} $p$ of $N$ and any multiple $aN$ of $N$.
It has been proved in [1] that the Riemann Hypothesis^{} is true iff 4 is the only extraordinary number. The proof is based on Gronwall’s theorem and Robin’s theorem.
References
- 1 Geoffrey Caveney, Jean-Louis Nicolas, Jonathan Sondow: Robin’s theorem, primes, and a new elementary reformulation of the Riemann Hypothesis. $-$ Integers 11 (2011) article A33; available directly at http://arxiv.org/pdf/1110.5078.pdfarXiv.
Title | extraordinary number |
---|---|
Canonical name | ExtraordinaryNumber |
Date of creation | 2013-03-22 19:33:41 |
Last modified on | 2013-03-22 19:33:41 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 14 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 11M26 |
Classification | msc 11A25 |
Related topic | PropertiesOfXiFunction |