Localized factorizations of integers

Abstract

We determine the order of magnitude of H(k+1)(x, y, 2y), the number of integers n ⩽ x that are divisible by a product d1·…·dk with yi < di ⩽ 2yi, when the numbers log y1, …, log yk have the same order of magnitude and k ⩾ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k + 1)-dimensional multiplication table as well as how many distinct sums of k + 1 Farey fractions there are modulo 1.

Locations

• Proceedings of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View