On Chen's theorem over Piatetski-Shapiro type primes and almost-primes

Abstract

In this paper, we establish a new mean value theorem of Bombieri-Vinogradov type over Piatetski-Shapiro sequence. Namely, it is proved that for any given constant $A>0$ and any sufficiently small $\epsilon >0$, there holds \begin{equation*} \sum_{\substack{d\leqslant x^ξ\\ (d,l)=1}}\Bigg|\sum_{\substack{A_1(x)\leqslant a<A_2(x)\\ (a,d)=1}}g(a) \Bigg(\sum_{\substack{ap\leqslant x\\ ap\equiv l\!\pmod d \\ ap=[k^{1/γ}]}}1 -\frac{1}{φ(d)}\sum_{\substack{ap\leqslant x\\ ap=[k^{1/γ}] }} 1\Bigg)\Bigg|\ll\frac{x^γ}{(\log x)^A}, \end{equation*} provided that $1⩽A_1(x)<A_2(x)⩽x^{1-\epsilon }$ and $g(a)\ll {\tau }_r^s(a)$, where $l\ne 0$ is a fixed integer and \begin{equation*} ξ:=ξ(γ)=\frac{2^{38}+17}{38}γ-\frac{2^{38}-1}{38}-\varepsilon \end{equation*} with \begin{equation*} 1-\frac{18}{2^{38}+17}<γ<1. \end{equation*} Moreover, for $\gamma$ satisfying \begin{equation*} 1-\frac{0.03208}{2^{38}+17}<γ<1, \end{equation*} we prove that there exist infinitely many primes $p$ such that $p+2={\mathcal{P}}_2$ with ${\mathcal{P}}_2$ being Piatetski-Shapiro almost-primes of type $\gamma$, and there exist infinitely many Piatetski-Shapiro primes $p$ of type $\gamma$ such that $p+2={\mathcal{P}}_2$. These results generalize the result of Pan and Ding [37] and constitutes an improvement upon a series of previous results of [29,31,39,47].

28 pages. arXiv admin note: substantial text overlap with arXiv:2003.04197