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arXiv · 2608.13299

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

Zhiyuan Yang
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Abstract

In this paper, we consider the asymptotic density of $\#\{p\leq x:P^+(p-1)\geq p^c\}$ and $\#\{n\leq x:P^+(n) 0.299x. \end{align*} The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq \frac{7}{2}\log\frac{1}{c}$. 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.

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Zhiyuan Yang. 2026-08-13. Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications. https://arxiv.org/abs/2608.13299

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