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’s type over Piatetski–Shapiro sequence. Namely, it is proved that for any given constant \(A>0\) and any sufficiently small \(\varepsilon >0\), there holds

$$\begin{aligned} \sum _{\begin{array}{c} d\leqslant x^\xi \\ (d,l)=1 \end{array}}\Bigg |\sum _{\begin{array}{c} A_1(x)\leqslant a<A_2(x)\\ (a,d)=1 \end{array}}g(a) \Bigg (\sum _{\begin{array}{c} ap\leqslant x\\ ap\equiv l\!\!\!\!\!\pmod d\\ ap=[k^{1/\gamma }] \end{array}}1-\frac{1}{\varphi (d)}\sum _{\begin{array}{c} ap\leqslant x\\ ap=[k^{1/\gamma }] \end{array}} 1\Bigg )\Bigg |\ll \frac{x^\gamma }{(\log x)^A}, \end{aligned}$$

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

$$\begin{aligned} \xi :=\xi (\gamma )=\frac{2^{38}+17}{38}\gamma -\frac{2^{38}-1}{38}-\varepsilon \end{aligned}$$

with

$$\begin{aligned} 1-\frac{18}{2^{38}+17}<\gamma <1. \end{aligned}$$

Moreover, for \(\gamma \) satisfying

$$\begin{aligned} 1-\frac{0.03208}{2^{38}+17}<\gamma <1, \end{aligned}$$

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 constitute an improvement upon a series of previous results of [29, 31, 39, 47].