This question is a follow up from About Goldbach's conjecture and comes from what I read about the Farey series related criterion for RH. Let $r_{k}(n)$ be the $k+1$-th potential typical primality radius of $n$. Based on the fact that $\mathcal{N}_{n}(x)\sim \dfrac{N_{1}(n)}{P_{ord_{C}(n)}}x$, where $\mathcal{N}_{n}(x)$ is the number of potential typical primality radii of $n$ below $x$ (see A possible consequence of Dirichlet's theorem about primes in arithmetic progression), one can expect $r_{k}(n)$ to be approximately equal to $(k+\frac{1}{2})\frac{P_{ord_{C}(n)}}{N_{1}(n)}$. Moreover, the heuristics based upper bound $\alpha_{n}=O(\sqrt{n}\log^{2}n)$ seems to imply that $\mathcal{N}_{n}(x)=\dfrac{N_{1}(n)}{P_{ord_{C}(n)}}x+O_{\varepsilon}(x^{\frac{1}{2}+\varepsilon})$.
So my question is:
is RH actually equivalent to $\displaystyle{\sum_{k=0}^{N_{1}(n)-1}\vert r_{k}(n)-(k+\frac{1}{2})\frac{P_{ord_{C}(n)}}{N_{1}(n)}\vert=O_{\varepsilon}(n^{\frac{1}{2}+\varepsilon})}$?
Thanks in advance.
