Primes in short arithmetic progressions

Abstract

Let x, h and Q be three parameters. We show that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, every reduced arithmetic progression a( mod q) has approximately the expected number of primes p from the interval (y, y + h], provided that h > x1/6+ϵ and Q satisfies appropriate bounds in terms of h and x. Moreover, we prove that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, there is at least one prime p ∈ (y, y + h] lying in every reduced arithmetic progression a( mod q), provided that 1 ≤ Q2 ≤ h/x1/15+ϵ.