The Barban–Davenport–Halberstam theorem for a restricted set of arithmetic progressions

Abstract

We consider the asymptotic version of the Barban–Davenport–Halberstam theorem in the form proved first by Hooley. Instead of averaging the expected error term over all reduced residue classes to modules in a given range, here we only consider those reduced arithmetic progressions that satisfy an additional congruence condition and provide an asymptotic approximation.