Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications

Abstract: In 1986, Bombieri, Friedlander and Iwaniec famously obtained that primes are equidistributed in arithmetic progressions to moduli up to $x<sup>{4/7-\varepsilon}$, using well-factorable weights. In this paper, we apply Pascadi's triply-well-factorable convolution estimate and his estimation of incomplete Kloosterman sums to generalize this result to a convolutionform, which improves Wang's result under certain conditions. As for application, we consider the asymptotic density of $#{n\leq x:P<sup>+(n)&lt;P<sup>+(n+1)}$ and $#{p\leq x:P^+(p-1)\geq p<sup>c}$, where $P<sup>+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, one has\begin{align*} #{n\leq x:P^+(n)<P^+(n+1)\}\>0.296x \end{align*} and \begin{align*} &\mathop{\lim\sup}{x\rightarrow\infty} \frac{1}{π(x)}#{p\leq x:P^+(p-1)\geq p^c}\&\leq S(c)=\left{ \begin{aligned} & \int_0^{1-c}\frac{2}{(5/8-189u/200)(1-u)}\mathrm{d}u, \quad&& \frac{184}{189}\leq c<1,\& S\left(\frac{184}{189}\right)+\frac{10}{3}\log\frac{184}{189c},\quad&& 0.7404<c\leq \frac{184}{189}, \end{aligned}\right. \end{align*} where the function $S(c)$ satisfties $S(c)&lt;\frac{7}{2}\log\frac{1}{c} $ for $0.7404<c<1$. The first result improves a previous result $0.280$ by the author (2026). The second result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\mathop{\lim\sup}</em>{x\rightarrow\infty} \frac{1}{π(x)}#{p\leq x:P^+(p-1)\geq p^c}\leq \frac{7}{2}\log\frac{1}{c}$.