What would be the consequences of proving Riemann's hypothesis for Legendre's conjecture?

Question

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?

Answer 1

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.

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