Primes in the form $[αp+β]$

Abstract

Let βbe a real number. Then for almost all irrational α>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}π_{α,β}^*(x)(\log x)^2/x>=1, where π_{α,β}^*(x)={p<=x: both p and [αp+β] are primes}.