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Harald Cramér proved that under this assumption that the Riemann hypothesis is true., the gap $g_n$ satisfies $$g_n = O(\sqrt{p_n} \ln p_n) ,$$

using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that $$g_n = O\left((\ln p_n)^2\right).$$

Would the Riemann hypothesis be true if the $(\ln p_n)^2$ conjecture was proven true? (I know the proof will not happens soon. But, if it does?)

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I do not believe that a proof of the Cramér conjecture would imply RH.

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  • $\begingroup$ Don't believe, why? What is the disconnect? $\endgroup$ – user160140 Apr 13 '14 at 12:08
  • $\begingroup$ @user160140: RH does imply somewhat short gaps between primes, but it requires that they are in a certain place (with regard to Li). Cramér says that the gaps are very short but doesn't care where they are. So although related, they're not obviously implicational. Of course the biggest piece of information is simply that, if such a result were known, it would be famous enough that I would know it. :) $\endgroup$ – Charles Apr 13 '14 at 16:27
  • $\begingroup$ So, I am taking a guess here. RH has implications with both size of gap and location of primes? $\endgroup$ – user160140 Apr 13 '14 at 19:14
  • $\begingroup$ @user160140: Yes. The primes must be within $O(\sqrt x\log x)$ of Li(x) and the gaps, consequently, can't be larger than $O(\sqrt x\log x)$. $\endgroup$ – Charles Apr 13 '14 at 19:38
  • $\begingroup$ So the RH shows were on Li(x) within $O(\sqrt x\log x)$, which is $O(\sqrt x\log x)$ < $O((\log x)^2$, and there is not a inverse of Li(x) as to find where the zeros are on the $i$-plane? $\endgroup$ – user160140 Apr 13 '14 at 19:55

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