This question is a follow-up to both About Goldbach's conjecture and Question in Proof of Hardy Ramanujan theorem about $\omega(n) =\sum_{p|n} 1$.

Can one derive from Robin's criterion for RH an inequality of the form $\sigma(n)\ll n\omega(n)$? Moreover, as Carl Pomerance told me in a private communication that the quantity $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ is expected to fulfill $r_{0}(n)\ll\log^{2}n\log\log n$, can one get from this criterion an upper bound for $r_{0}(n)$ of the form $r_{0}(n)\ll\frac{\sigma(n)}{N_{2}(n)}$ where $N_{2}(n)=\sharp\{0<r<n-\sqrt{2n-3},(n-r,n+r)\in\mathbb{P}^{2}\}$ under RH? Finally, whenever $n$ is a perfect number, does $\sigma(n)=2n$ imply (always assuming RH) that $r_{0}(n)<C\log^{2}n$ for some positive constant $C$?