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?
Would Legendre's Conjecture imply Riemann's Hypothesis?
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prime-numbers
riemann-zeta
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1This question has [an answer there](http://math.stackexchange.com/a/482845/276986). The consequence of the RH on the prime gap [are not easy](http://matwbn.icm.edu.pl/ksiazki/aa/aa41/aa4118.pdf). – 2017-01-22
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0That's right, I did not found that information. So sorry – 2017-01-22