# Is RH equivalent to the following estimate?

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.

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I'm curious too, this $\tfrac{1}{2}+\varepsilon$ keeps coming up again and again.... – Suvrit Oct 31 '13 at 18:35
"potential typical primality radius" It appears that again you are speaking your own private language. – Gerry Myerson Oct 31 '13 at 22:35
I suppose that once again I have to copy-paste the content of the first link? – Sylvain JULIEN Oct 31 '13 at 22:41