The distribution of integers with a divisor in a given interval

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 $z\le x^{1/2-\varepsilon}$. For every $r\ge 2$, $C>1$ and $\varepsilon>0$, we determine the order of magnitude of $H_r(x,y,z)$ uniformly for $y$ large and $y+y/(\log y)^{\log 4 -1 – \varepsilon} \le z \le \min(y^{C},x^{1/2-\varepsilon})$. As a consequence of these bounds, we settle a 1960 conjecture of Erdős and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.