Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications
Abstract
In this paper, we consider the asymptotic density of #{p≤ x:P+(p−1)≥ pc} and #{n≤ x:P+(n)<P+(n+1)}, where P+(n) denote the largest prime factor of n. We show that for x→∞, one has mathoplimsupx→∞ 1/π(x)#{p≤ x:P+(p−1)≥ pc}≤16/5log1/c where e−5/16<c<1, and #{n≤ x:P+(n)<P+(n+1)}>0.299x. The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined mathoplimsupx→∞ 1/π(x)#{p≤ x:P+(p−1)≥ pc}≤ 7/2log1/c. The second result improves a previous result $0.280$ by the author (2026). The key to the proof is that for a special class of convolution forms equipped with well-factorable weights, we may use the level x5/8−o(1) for Pascadi's prime-distribution result with triple-well-factorable weights. We also use Pascadi's estimation of incomplete Kloosterman sums.
Related
- A Bombieri-Vinogradov theorem for exponential sums over products of k primes
- Eberlein convolution as an inner product and quantitative Bombieri--Taylor results
- A Second-Order Extension of Hájek's Convolution Theorem with Statistical Applications
- Applying the convolution theorem to the two-dimensional magnetic inverse problem
- On the convolution theorem for directionally pathwise differentiable functionals
BibTeX
@misc{yang2026convolutiontype,
title = {{Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications}},
author = {Zhiyuan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13299},
eprint = {2608.13299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13299},
}
RIS
TY - UNPB
TI - Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications
AU - Zhiyuan Yang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.13299
UR - https://arxiv.org/abs/2608.13299
AB - In this paper, we consider the asymptotic density of #{p≤ x:P^+(p−1)≥ p^c} and #{n≤ x:P^+(n)<P^+(n+1)}, where P^+(n) denote the largest prime factor of n. We show that for x→∞, one has mathoplimsup_x→∞ 1/π(x)#{p≤ x:P^+(p−1)≥ p^c}≤16/5log1/c where e^−5/16<c<1, and #{n≤ x:P^+(n)<P^+(n+1)}>0.299x. The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined mathoplimsup_x→∞ 1/π(x)#{p≤ x:P^+(p−1)≥ p^c}≤ 7/2log1/c. The second result improves a previous result $0.280$ by the author (2026). The key to the proof is that for a special class of convolution forms equipped with well-factorable weights, we may use the level x^5/8−o(1) for Pascadi's prime-distribution result with triple-well-factorable weights. We also use Pascadi's estimation of incomplete Kloosterman sums.
ER -