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I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus $1\ll\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}\ll\log^{2} x$ with current technology unconditionally, or if we need rather strong conjectures like (G)RH and/or the Generalized Elliott-Halberstam conjecture. Has such a question been investigated so far? If so, are there interesting references to read?

Many thanks in advance.

Edited after the second comment below: Does RH imply that, if $\beta:=\inf\{c,\limsup \dfrac{p_{n+1}-p_{n}}{\log^{c} p_{n}}=O(1)\}$ then $\beta=2$ ?

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    $\begingroup$ I think I do not quite get the question, $\pi(x+r(x))-\pi(x-r(x))$ Could just be $0$ if the interval does not contain any prime. And "of course" we cannot prove the existence of primes in as short intervals as $(\log(x))^4$. This is way beyond current results. $\endgroup$
    – user9072
    Commented Oct 10, 2015 at 17:31
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    $\begingroup$ Using RH with $r(x) = \log^2(x)$, Selberg proved that for most $x$ there is the right asymptotic, namely $\frac{r(x)}{\pi(x+r(x))-\pi(x-r(x))} \sim 2\log(x)$. Here most means for all but a set of asymptotic density zero. $\endgroup$
    – Lior Bary-Soroker
    Commented Oct 10, 2015 at 17:37
  • $\begingroup$ That's extremely interesting! The quantity $r(x)$ I consider is "on average" of the order of $\dfrac{1}{2}\log^{2}(x)$, so that would suit pretty well my question. Would you have a link to this paper of Selberg? $\endgroup$
    – Sylvain JULIEN
    Commented Oct 10, 2015 at 17:40

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