# If Cramér's is proved?

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?)

• @user160140: Yes. The primes must be within $O(\sqrt x\log x)$ of Li(x) and the gaps, consequently, can't be larger than $O(\sqrt x\log x)$. – Charles Apr 13 '14 at 19:38
• 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? – user160140 Apr 13 '14 at 19:55