An Additive Problem over Piatetski-Shapiro Primes and Almost-primes

Abstract

Let ${\mathcal{P}}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/\gamma }]$ with $\frac{85}{86}<\gamma <1$. Moreover, we use this result to prove that, for $0.9989445<\gamma <1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2={\mathcal{P}}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.

29 pages