Bombieri-Vinogradov in short intervals

Question

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the reduction of the lower bound on $\theta$?

I'm not fussed if some of the other variables are changed as, clearly, $\psi$ would have to be altered if we could get $\theta < 1/2$, but I just need a good $\theta$ whilst keeping (roughly) the same R.H.S..

Answer 1

As far as I know the Prime Number Theorem in the form $\pi(x+x^\theta)-\pi(x)\sim\frac{x^\theta}{\log x}$ is not proven for any fixed $\theta<\frac{7}{12}$. So I guess the best you could hope for would be $\theta=\frac{7}{12}-\omega(x)$, where $\omega\searrow0$, but even that would be pretty difficult.