The distribution of integers with a divisor in a given interval
by
Kevin Ford
University of Illinois at Urbana-Champaign

We investigate the growth rate of the functions H(x,y,z), which counts the number of integers up to x that have a divisors in (y,z], and H_r(x,y,z), which counts the number of integers up to x which have exactly r divisors in (y,z]. We determine the order of magnitude of H(x,y,z) for all x,y,z and, for all r, the order of magnitude of H_r(x,y,z) for a large set of x,y,z including the important special case z=2y. Using these bounds we settle conjectures of Erdos and Tenenbaum. An important ingredient is a new probability bound for uniform order statistics.

Date received: April 12, 2004