The distribution of integers with a divisor in a given interval

We determine the order of magnitude of H(x, y, z), the number of integers n 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 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 x 1/2- . For every r 2, C > 1 and > 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- z min(y C , x 1/2- ). As a consequence of these bounds, we settle a 1960 conjecture of Erds and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.