Primes in short arithmetic progressions

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mathrm{m}\mathrm{o}\mathrm{d}q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1\mathrm{/}6+ϵ}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least one prime $p\in (y,y+h]$ lying in every reduced arithmetic progression $a\mod q$,providedthat$1\le Q^2\le h/x^{1/15+\epsilon}$.