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# 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

Major Section:

Reference

Type of Math Object:

Definition

Parent:

## Mathematics Subject Classification

11M26*no label found*11A25

*no label found*

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new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier