In 1985 [Perelli, Pintz & Salerno][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..


  [1]: http://link.springer.com/article/10.1007%2FBF01388653#page-1