发表机构
School of Mathematics, Shandong University（山东大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将带良因子可分权重的Bombieri-Vinogradov定理推广为卷积形式，改进了Wang的相关结果，并在最大素因子相关的素数分布问题中得到了更优的渐近密度上界。
AI 中文摘要
1986年，Bombieri、Friedlander和Iwaniec利用良因子可分权重，著名地证明了素数在模不超过$x^{4/7-\varepsilon}$的算术级数中是等分布的。本文中，我们应用Pascadi的三重良因子可分卷积估计及其不完全Kloosterman和的估计，将该结果推广为卷积形式，在特定条件下改进了Wang的结果。作为应用，我们考虑$\\#\{n\leq x:P^+(n)<P^+(n+1)\}$和$\\#\{p\leq x:P^+(p-1)\geq p^c\}$的渐近密度，其中$P^+(n)$表示$n$的最大素因子。我们证明当$x\rightarrow\infty$时，$\\#\{n\leq x:P^+(n)<P^+(n+1)\}>0.296x$；且$\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\\#\{p\leq x:P^+(p-1)\geq p^c\}\leq S(c)$，其中$S(c)$满足分段表达式，且对$0.7404<c<1$有$S(c)<\frac{7}{2}\log\frac{1}{c}$。第一个结果改进了作者（2026）之前的结果0.280，第二个结果改进了Ding和Wang（2025）的结果，后者得到的上界为$\frac{7}{2}\log\frac{1}{c}$。
英文摘要
In this paper, we consider the asymptotic density of $\#\{p\leq x:P^+(p-1)\geq p^c\}$ and $\#\{n\leq x:P^+(n)<P^+(n+1)\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, one has \begin{align*} \mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{π(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq\frac{16}{5}\log\frac{1}{c} \end{align*} where $e^{-5/16}<c<1$, and \begin{align*} \#\{n\leq x:P^+(n)<P^+(n+1)\}>0.299x. \end{align*} The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{π(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq \frac{7}{2}\log\frac{1}{c}$. 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.
Comments27 pages
