The distribution of integers with a divisor in a given interval

TL;DR

This paper determines the order of magnitude of the count of integers up to x with divisors in a given interval, resolving a 1960 conjecture of Erdős and related problems using new results on order statistics.

Contribution

It provides the first comprehensive order of magnitude results for integers with divisors in intervals, including a resolution of Erdős's 1960 conjecture.

Findings

Established the order of magnitude of H(x,y,z) for all x,y,z.

Determined the order of magnitude of H_r(x,y,z) for r=1 and r≥2 under various conditions.

Settled Erdős's 1960 conjecture and related conjectures.

Abstract

We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le \min(y^{C},x^{1/2-\epsilon}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics.