Timeline for Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

6 events

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Oct 11, 2015 at 10:16	history	edited	Sylvain JULIEN	CC BY-SA 3.0	added 152 characters in body
Oct 10, 2015 at 17:40	comment	added	Sylvain JULIEN		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?
Oct 10, 2015 at 17:37	comment	added	Lior Bary-Soroker		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.
Oct 10, 2015 at 17:31	comment	added	user9072		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.
Oct 10, 2015 at 17:26	history	edited	Sylvain JULIEN	CC BY-SA 3.0	added 3 characters in body
Oct 10, 2015 at 17:15	history	asked	Sylvain JULIEN	CC BY-SA 3.0