# extraordinary number

Define the function $G$ for integers $n>1$ by

 $G(n)\;:=\;\frac{\sigma(n)}{n\ln(\ln{n})},$

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

 $G(N)\;\geq\;\max\{G(N/p),\,G(aN)\}$

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 ExtraordinaryNumber 2013-03-22 19:33:41 2013-03-22 19:33:41 pahio (2872) pahio (2872) 14 pahio (2872) Definition msc 11M26 msc 11A25 PropertiesOfXiFunction