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

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

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.

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