In 1985 [Perelli, Pintz & Salerno](http://link.springer.com/article/10.1007%2FBF01388653#page-1) 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..



Bombieri-Vinogradov in short intervals
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.


