A lower bound involving the divisor function and primorial numbers It is known that
$\lim$ $\sup \dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_k}=\frac{6}{\pi^2}$,
where $\gamma$ is the Euler-Mascheroni constant and $N_k$ is the $k-th$ primorial number.
But is it true that, for sufficiently large $N_k$, 
$ \dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_k} >\frac{6}{\pi^2}$ ?
 A: The affirmative answer to your question is equivalent to the Riemann Hypothesis, and this was observed by Solé and Planat as a consequence of Nicolas's earlier work. I recall their argument briefly.
Assume that the Riemann Hypothesis holds. Then Nicolas's Theorem 2 (a) implies readily that $\frac{\sigma(N_k)}{N_k \log\log N_k} >\frac{6}{\pi^2}e^\gamma$ holds for all $k$. Now assume that the Riemann Hypothesis fails. Then Nicolas's Theorem 3 (c) implies readily that the inequality holds for infinitely many $k$'s, and also that the inequality fails for infinitely many $k$'s.
A: From Jean-Louis Nicolas, http://math.univ-lyon1.fr/~nicolas/ in  article we find that there are infinitely many primorials $n$ such that
$$ \frac{1}{e^\gamma \log \log n} > \frac{\phi(n)}{n} $$
Therefore, infinitely often,
$$ \frac{1}{e^\gamma \log \log n}  \frac{\sigma(n)}{n}> \frac{\phi(n)}{n} \frac{\sigma(n)}{n} =  \frac{\sigma(n) \phi(n)}{n^2} > \frac{6}{\pi^2} $$
from Theorem 329 and footnote on page 267 of Hardy and Wright.
I am unable to tell whether your inequality for sufficiently large primorials is equivalent to RH. You are mixing two types of behavior, extreme behavior for primorials is the Nicolas criterion, while that for colossally abundant numbers is the Robin criterion. The inequalities in Hardy and Wright linking $\phi(n)$ and $\sigma(n)$ are not tight enough to cross the conditions.  
