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.

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 $\beta=2$ ?

I would like to know whether it is possible to obtain the bounds $\sqrt{x}\ll k(x)\ll r(x)$ 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.

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.