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. It is known that the Riemann hypothesis is true if and only if there exist infinitely many extremely abundant numbers. In this note, using this criterion on hyper abundant numbers, we prove that the Riemann hypothesis is true.