TITLE: What would be the consequences of proving Riemann's hypothesis for Legendre's conjecture?
QUESTION [5 upvotes]: I've heard somewhere that Riemann's hypothesis doesn't imply Legendre's conjecture. But if Riemann's hypothesis is true, would an interval maybe a bit larger than $[n^2,(n+1)^2]$ contain always at least one prime?

REPLY [2 votes]: Yes. In particular, Cramér proved that under RH, prime gaps are $O(\sqrt x\log x)$ and this can be made effective with a result of Schoenfeld. So you're 'only' off by a factor of roughly $\frac{\log x}{4\pi}$ once you have RH.

Harold Cramér, Some theorems concerning prime numbers, Arkiv för Matematik, Astronomi och Fysik 15, #5 (1920), pp. 1–32.
Lowell Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II, Mathematics of Computation 30:134 (1976), pp. 337–360.