Riemann Hypothesis on Grönwall's Function

Grönwall's function $G$ is defined for all natural numbers $n>1$ by $G(n)=\frac{\sigma(n)}{n \cdot \log \log n}$ where $\sigma(n)$ is the sum of the divisors of $n$ and $\log$ is the natural logarithm. We require the properties of extremely abundant numbers, that is to say left to right maxima of $n \mapsto G(n)$. We also use the colossally abundant and hyper abundant numbers. There are several statements equivalent to the famous Riemann hypothesis. We state that the Riemann hypothesis is true if and only if there exist infinitely many pairs $(N,N')$ of consecutive colossally abundant numbers $N< N'$ such that $G(N)< G(N')$. Using this new criterion, we prove that the Riemann hypothesis is true.