On the distribution of $\alpha p^2$ modulo one with prime $p$ of a special form

Fei Xue; Jinjiang Li; Min Zhang

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arXiv · 2103.11621

On the distribution of $\alpha p^2$ modulo one with prime $p$ of a special form

Fei Xue·Jinjiang Li·Min Zhang

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Abstract

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there exist infinitely many primes $p$, such that $\|\alpha p^2+\beta\|<p^{-\theta}$ and $p+2=\mathcal{P}_4$, which constitutes an improvement upon the previous result.

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Fei Xue, Jinjiang Li, Min Zhang. 2021-03-22. On the distribution of $\alpha p^2$ modulo one with prime $p$ of a special form. https://arxiv.org/abs/2103.11621

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