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

Abstract

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

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