On the A-dependence of the linear sieve with application to polynomial sequences

AbstractLet A denote a finite sequence of integers and put Ad = {a ∈ A : a ≡ 0(d)}. Let P denote a set of distinct primes and write P(z) = Πp < z, p ∈ P p, z ≥ 2. Assume that there exists a convenient approximation X to |A| and a nonnegative, multiplicative, arithmetic function ω(d) on the divisors d of P(z), such that the remainder Rd := |Ad| − (ω(d)d)X are small on average over all divisors d of P(z) that are less than a certain number y. Jurkat and Richert introduced the well-known functions f(u) and F(u) to show that under the linear sieve condition Πv⩽p<w1−w(p)p−1⩽logwlogv1+Alogv, 2⩽v<w, (1) the upper and lower sieve bounds S(A,P,z):= Σaϵ,A,(a,P(z))=1 1⪋X Πp<z,pϵP1−w(p)PxF(u)+f(u)−H+−∑d