Convergent sieve sequences in arithmetic progressions

Let V be a set of pairwise coprime integers not containing 1 and suppose, there is a 0⩽δ<1, such that ∑v∈Vv−1+δ<∞ holds. Let χV(n)=1 if v∤n for all v∈V and χV(n)=0 elsewhere. We study the behavior of χV in arithmetic progressions uniformly in the modulus, both individually and in the quadratic mean over the residue classes. As an application, new bounds for the mean square error of squarefree numbers in arithmetic progressions are obtained.