Primes in Short Segments of Arithmetic Progressions

Λ is the von Mangoldt function, and ∑ * a(q) denotes a sum over a set of reduced residues modulo q. We shall assume throughout x ≥ 2, 1 ≤ q ≤ x, 1 ≤ h ≤ x, (1.3) the other ranges being without interest. As far as we are aware the only known result concerning the general function I(x, h, q) is due to Prachar [11], who showed that, assuming the Generalized Riemann Hypothesis (GRH) I(x, h, q) ≪ hx log qx. (1.4) On the other hand, much more is known about the special cases where one of the two aspects, segment or progression, is trivialized. Indeed, our function I(x, h, q) is essentially a hybrid of the more familiar functions