Localized factorizations of integers

TL;DR

This paper analyzes the distribution of integers with specific factorizations within certain bounds, generalizing previous results and applying these bounds to problems involving multiplication tables and Farey fractions.

Contribution

It extends the understanding of the distribution of integers with factorizations to multiple dimensions, generalizing Ford's earlier work for k=1.

Findings

Derived bounds for the number of integers with specified factorizations in multiple dimensions.

Connected these bounds to the count of elements in multiplication tables.

Applied results to the distribution of sums of Farey fractions modulo 1.

Abstract

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\le 2y_i, when the numbers \log y_1,...,\log y_k have the same order of magnitude and k\ge 2. This generalizes a result by K. 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.