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 $\varepsilon>0$, there holds \begin{equation*} \sum_{\substack{d\leqslant x^\xi\\ (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/\gamma}]}}1 -\frac{1}{\varphi(d)}\sum_{\substack{ap\leqslant x\\ ap=[k^{1/\gamma}] }} 1\Bigg)\Bigg|\ll\frac{x^\gamma}{(\log x)^A}, \end{equation*} provided that $1\leqslant A_1(x)<A_2(x)\leqslant x^{1-\varepsilon}$ and $g(a)\ll \tau_r^s(a)$, where $l\not=0$ is a fixed integer and \begin{equation*} \xi:=\xi(\gamma)=\frac{2^{38}+17}{38}\gamma-\frac{2^{38}-1}{38}-\varepsilon \end{equation*} with \begin{equation*} 1-\frac{18}{2^{38}+17}<\gamma<1. \end{equation*} Moreover, for $\gamma$ satisfying \begin{equation*} 1-\frac{0.03208}{2^{38}+17}<\gamma<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].