{"paper":{"title":"Localized factorizations of integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dimitris Koukoulopoulos","submitted_at":"2008-09-05T16:19:16Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.1072","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}