A Sieve Problem and Its Application

Abstract

Let θ be an arithmetic function and let MJX-tex-caligraphicscriptB be the set of positive integers n=p1α1⋯pkαk which satisfy pj+1⩽θ(p1α1⋯pjαj) for 0⩽j<k. We show that MJX-tex-caligraphicscriptB has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of MJX-tex-caligraphicscriptB. When θ(n)/n is non‐decreasing, the set MJX-tex-caligraphicscriptB coincides with the set of integers n whose divisors 1=d1<d2<⋯<dτ(n)=n satisfy dj+1⩽θ(dj) for 1⩽j<τ(n).

The number of prime factors of integers with dense divisors

“…If B(x) = o(x), the validity of (31) at s = 1 follows from (28), in much the same way that the validity of (30) at s = 1 follows from (27), which is demonstrated in the proof of [15,Thm. 1].…”

“…If q < pt, this is implied by T ≪ c p / log q, which follows from (17) (with q replaced by p). If q ≥ pt, we estimate λ n with Mertens' formula and use Abel summation and the estimate (15). The contribution from the first two error terms in ( 15) is clearly acceptable, while the term O(1) contributes ≪ ∞ q/t dy y 2 log yt ≤ t q log q ≍ c p log q…”

An extension of the Siegel-Walfisz theorem

“…If θ(n) ≥ n, then R(x, q) < P + (q). With Lemma 1, the known asymptotic results for B θ (x) under various conditions on θ(n) (see [12,13]), applied with θ q in place of θ, translate to asymptotics for B θ,q (x), which in turn lead to estimates for B(x, q, a), by Corollary 2. Since the implied constants in the error terms of these estimates depend on θ, any estimates for B(x, q, a) derived in this manner will have implied constants that depend on q.…”

The mean number of divisors for rough, dense and practical numbers

“…The improvement comes from a more careful treatment of the penultimate term in Lemma 2, when this estimate is inserted into Lemma 4. If B(x) ∼ c θ x/ log x, then θ(n) cannot grow much faster than is allowed by Theorem 4, since θ(n) n 1+ε implies B(x) ∼ c θ x/(log x) 1−η for some η > 0, by [14,Thm. 3].…”