PreprintCC0arXiv
On integers whose friable component is large
Abstract
We estimate the number of integers up to x whose y-friable component exceeds z. Uniformly for max{2,log z}leqslant yleqslant zleqslant x/2, we determine the logarithm of this counting function up to an error O(log z/log y). This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friable-number estimates with an elementary argument for the lower bound.
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BibTeX
@misc{tenenbaum2026integers,
title = {{On integers whose friable component is large}},
author = {Gérald Tenenbaum and Xiangyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37507},
eprint = {2609.37507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37507},
note = {CC0},
}
RIS
TY - UNPB
TI - On integers whose friable component is large
AU - Gérald Tenenbaum
AU - Xiangyu Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.37507
UR - https://arxiv.org/abs/2609.37507
AB - We estimate the number of integers up to x whose y-friable component exceeds z. Uniformly for max{2,log z}leqslant yleqslant zleqslant x/2, we determine the logarithm of this counting function up to an error O(log z/log y). This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friable-number estimates with an elementary argument for the lower bound.
ER -