# Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$.

I call $r$ a primality radius of $n$, $r_{0}(n)$ the fundamental primality radius of $n$ and $k_{0}(n)$ the order of centrality of $n$. I say that $n$ is a $k$-central number if and only if $k_{0}(n)=k$.

Now, the number of $k$-central numbers less than $x$ $\pi_{C,k}(x)$ should verify the following relation:

$$\pi_{C,k}(x)\sim \vert\{n\le x, k_{0}(n)=k\}\vert$$

and thus $\pi_{C,k}(x)\le\dfrac{\pi(x+\max_{n\le x}r_{0}(n))}{k}(1+o(1))$.

I now formulate the following conjecture:

Negligible fundamental primality radius conjecture (NFPR conjecture for short):

$$\forall\varepsilon>0,\forall x>2, \max_{n\le x}r_{0}(n)=O_{\varepsilon}(x^{\varepsilon})$$

Could one deduce from this conjecture that $\dfrac{\pi(x+\max_{n\le x}r_{0}(n))}{k}\sim\dfrac{\pi(x)}{k}$?

If so, one would have $\pi_{C,k}(x)\le \dfrac{\pi(x)}{k}(1+o(1))$.

Hence $\displaystyle{N_{k}(x):=\sum_{l=0}^{k}\pi_{C,l}(x)\le\pi(x)(1+H_{k})(1+o(1))}$, where $H_{k}$ is the $k$-th harmonic number.

So that one should have $N_{k}(p_{n+k})-N_{k}(p_{n})\le k(1+H_{k})(1+o(1))=O(k\log k)$.

Now, from the prime number theorem, $N_{k}(x)\sim x$ for $k$ large enough and less than $x$. So, is it possible to prove rigorously that the conjunction of Goldbach's conjecture and NFPR conjecture would entail that $\lim\inf_{n\to\infty} p_{n+k}-p_{n}=O(k\log k)$ which, as stated in http://arxiv.org/pdf/1306.0948.pdf, follows from Hardy-Littlewood's prime k-tuples conjecture?

EDIT November 22nd 2013: I guess a better way to give a estimation of $\pi_{C,k}(x)$, and thus of $N_{k}(x)$, would be to establish rigorously that $\pi_{C,k}(x)\asymp \frac{\pi(x)}{k}$ under NFPR conjecture (since the PNT shows that under this conjecture, $\pi(x+r_{0}(x))\sim\pi(x)$). Is such an asymptotics correct?