Convergent sieve sequences in arithmetic progressions

Abstract

Let V be a set of pairwise coprime integers not containing 1 and suppose, there is a 0 ⩽ δ < 1 , such that ∑ v ∈ V v − 1 + δ < ∞ holds. Let χ V ( n ) = 1 if v ∤ n for all v ∈ V and χ V ( n ) = 0 elsewhere. We study the behavior of χ V 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.