Are there any results showing a prime exists in $(n^2/4-n,n^2/4)$?

Question

I see interval proofs of a prime in $(2n,3n)$ and $(4n,5n)$ etc. all the time, and found one paper that showed it for up to $(519n,520n)$.

Are there results that show a prime in $(\frac{1}{4}n^2-n,\frac{1}{4}n^2)$ or anything similar? Or are the only proofs so far related in linear terms of $n$?

Answer 1

The precise thing you ask for is not known, but better than linear (and close to quadratic) is possible. The best thing we can prove at the moment is that for every large enough $x$, there is a prime in the interval $[x - x^{0.525}, x]$. See here. The Riemann Hypothesis would improve the exponent $0.525$ to $0.5 + \epsilon$.

Answer 2

Oppermann's conjecture for $(n^2-n, n^2)$ is still open.