Robin's criterion, Goldbach's conjecture and upper bound for $r_{0}(n)$

Question

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 $C$?

Answer 1

The asymptotic $\sigma(n)\ll n\omega(n)$ is true and has nothing to do with RH or Robin's criterion.

Let us rewrite it as $\frac{\sigma(n)}{n}\ll\omega(n)$. By writing $n=\prod_{i=1}^k p_i^{e_i}$, where $p_1<\dots<p_k$ are primes and $e_i>0$, we want to show $$\prod_{i=1}^k\frac{\sigma(p_{e_i}^{e_i})}{p_{e_i}^{e_i}}\ll k.$$ But $\frac{\sigma(p^k)}{p^k}=1+\frac{1}{p_i}+\dots+\frac{1}{p_i^{e_i}}<1+\frac{1}{p_i-1}\leq 1+\frac{1}{i}$, so $$\frac{\sigma(n)}{n}=\prod_{i=1}^k\frac{\sigma(p_{e_i}^{e_i})}{p_{e_i}^{e_i}}\leq\prod_{i=1}^k\left(1+\frac{1}{i}\right)=k+1.$$

I have no idea about the quantities related to Goldbach conjecture but I strongly doubt RH can prove anything like what you ask for.