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 $x^{4/7-\epsilon }$, 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 $\#\{n\le x:P^{+}(n)<P^{+}(n+1)\}$ and $\#\{p\le x:P^{+}(p-1)\ge p^c\}$, where $P^{+}(n)$ denote the largest prime factor of $n$. We show that for $x\to \infty$, 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 $S(c)<\frac{7}{2}\log \frac{1}{c}$ 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 ${\mathrm{limsup}}_{x\to \infty }\frac{1}{\pi (x)}\#\{p\le x:P^{+}(p-1)\ge p^c\}\le \frac{7}{2}\log \frac{1}{c}$.

29 pages