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.