Localized factorizations of integers
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.
Cite
@article{arxiv.0809.1072,
title = {Localized factorizations of integers},
author = {Dimitris Koukoulopoulos},
journal= {arXiv preprint arXiv:0809.1072},
year = {2013}
}
Comments
34 pages. Added reference [10] to a paper of Nair and Tenenbaum which contains a result similar to Lemma 2.2 and which appeared prior to the publication of this paper. Simplified the proof of Lemma 2.2 using ideas from [10]. Removed part (a) of Lemma 2.2, as it is now redundant. Version 5 remains the published version