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 H₁(x, y, z) for all x; y; z satisfying z ≤ x^1/2-ε. For every r ≥ 2, C > 1 and " ε > 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-ε ≤ z ≤ min(y^C, x^1/2-ε). As a consequence of these bounds, we settle a 1960 con-jecture of Erdos and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.