I'm reading Robin's paper, ''Grandes valeurs de la fonction somme des diviseurs et hypoth`ese de Riemann,'' J. Math. Pures Appl. (9) 63 (1984).
In particular, Lemma 5 states that
$\prod_{p\leq P} (1-p^{-1})^{-1} < e^{\gamma}\log \theta(P)\exp\Big(\frac{2+c}{\sqrt{P}\log P} + \frac{\alpha(P)}{\sqrt{P}\log^{2}P}\Big)$, where $\theta(x)=\sum_{p\leq x} \log p,$ $p$ denotes a prime and $\gamma$ is the Euler-Mascheroni constant. My understanding of French is not great, but it seems this result was proved unconditionally (more specifically without the RH). However, it seems to me that the RH would follow too easily from this.
I mean, from this result it can be shown that For every large enough colossally abundant (CA) number $N$, one has $\frac{\sigma(N)}{N}<e^{\gamma}\log \log N + \frac{5\log \log N}{\sqrt{\log N}\log \log N}$.
Indeed, let $N$ be an arbitrarily large CA number and let $p$ denote a prime. Define $P$ to be the largest prime factor of $N$. From the prime factorisation of $N$, it follows that \begin{equation} \theta(P):=\sum_{p \leq P} \log p \leq \log N \end{equation} The aforesaid Lemma 5 states that \begin{equation} \frac{\sigma(N)}{N}<\prod_{p\leq P} (1-p^{-1})^{-1} < e^{\gamma}\log \theta(P)\exp\Big(\frac{2+c}{\sqrt{P}\log P} + \frac{\alpha(P)}{\sqrt{P}\log^{2}P}\Big) \end{equation} where $c=0.04619$ and $\alpha(P)<3335$. It is well-known that $P \sim \log N$. Applying this together with the fact that $\theta(P)\leq \log N$ into the above yields \begin{equation} \frac{\sigma(N)}{N}<e^{\gamma}\log \log N + \frac{5\log \log N}{\sqrt{\log N}\log \log N}. \end{equation}
Now, suppose the RH is false. Then by Proposition 4 of the same paper of Robin , there should exist infinitely many CA numbers $n$ such that $\frac{\sigma(n)}{n\log \log n}=e^{\gamma} + \Omega_{\pm}((\log n)^{-C})$ for some constant $C \in (0, \frac{1}{2})$. But this would contradict the above inequality, thus we deduce that our supposition must be false thus the RH must be true ?
