Integers with a divisor in (y,2y]

Abstract

We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.

Upper bound simplified with a lemma of Koukoulopoulos (Lemma 3.3), for which we give a much simpler proof. A few small corrections. Will be updated if further improvements and/or simplifications to the arguments are found. v4 will remain the 'legacy' version corresponding to the published version