## You are here

Homeextraordinary number

## Primary tabs

# 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 arXiv.

Keywords:

sum of divisors, Riemann Hypothesis

Related:

PropertiesOfXiFunction

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

11M26*no label found*11A25

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag