Riemann's hypothesis implies that the gap $g_n$ between primes is $$g_n=O(\sqrt{p_n}\log{p_n})$$ And Legendre's conjecture is equivalent to $$g_n=O(\sqrt{p_n})$$
Then, would proving the second conjecture imply the Riemann's Hypothesis to be true?
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1This question has an answer there. The consequence of the RH on the prime gap are not easy. – reuns Jan 22 '17 at 00:20
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That's right, I did not found that information. So sorry – user3141592 Jan 22 '17 at 00:50