A sieve problem and its application

TL;DR

This paper investigates a specific sieve problem involving the set of integers with prime factors constrained by an arithmetic function, establishing its natural density, criteria for positivity, and estimates for its counting function.

Contribution

It introduces a new sieve set defined by a function-based prime factorization condition, analyzes its density, and provides criteria and estimates for its distribution.

Findings

The set has a well-defined natural density.

A criterion for the density to be positive is established.

Various estimates for the counting function are derived.

Abstract

Let $\theta$ be an arithmetic function and let $\mathcal{B}$ be the set of positive integers $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$, which satisfy $p_{j+1} \le \theta ( p_1^{\alpha_1}\cdots p_{j}^{\alpha_{j}})$ for $0\le j < k$. We show that $\mathcal{B}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of $\mathcal{B}$. When $\theta(n)/n$ is non-decreasing, the set $\mathcal{B}$ coincides with the set of integers $n$ whose divisors $1=d_1< d_2 < \ldots <d_{\tau(n)}=n$ satisfy $d_{j+1} \le \theta( d_j )$ for $1\le j <\tau(n)$.