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Smooth numbers in simultaneous arithmetic progressions
Abstract
We prove a mean value theorem on the distribution of smooth numbers in two simultaneous arithmetic progressions, with generalizing a result of Drappeau. Let ε>0 and (log x)C<y<x1/C for a sufficiently large C. Our result implies that y-smooth numbers are well-distributed in the residue classes of a fixed integer a on average over moduli ≤ x5/9−ε for each |a|≤ x, which improves Drappeau's range ≤ x6/11−ε. As an application, we show that a positive proportion of y-smooth numbers n≤ x satisfy min(P+(n−1), P+(n+1))>x407/720.
Related
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BibTeX
@misc{yang2026smooth,
title = {{Smooth numbers in simultaneous arithmetic progressions}},
author = {Zhiyuan Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01522},
eprint = {2611.01522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01522},
}
RIS
TY - UNPB
TI - Smooth numbers in simultaneous arithmetic progressions
AU - Zhiyuan Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01522
UR - https://arxiv.org/abs/2611.01522
AB - We prove a mean value theorem on the distribution of smooth numbers in two simultaneous arithmetic progressions, with generalizing a result of Drappeau. Let ε>0 and (log x)^C<y<x^1/C for a sufficiently large C. Our result implies that y-smooth numbers are well-distributed in the residue classes of a fixed integer a on average over moduli ≤ x^5/9−ε for each |a|≤ x, which improves Drappeau's range ≤ x^6/11−ε. As an application, we show that a positive proportion of y-smooth numbers n≤ x satisfy min(P^+(n−1), P^+(n+1))>x^407/720.
ER -