Primes in the form $[\alpha p+\beta]$

TL;DR

This paper investigates the distribution of primes p for which both p and the integer part of lpha p + eta are prime, establishing a limit superior result for almost all irrational lpha > 0.

Contribution

It proves a new asymptotic lower bound for the density of such primes in the context of almost all irrational lpha, extending prime distribution theory.

Findings

or almost all irrational lpha > 0, the limsup of ount of primes p with both p and [lpha p + eta] prime grows at least as fast as x/( ( )^2].

stablishes a measure-theoretic result linking irrational lpha and prime pairs.

ontributes to understanding the joint distribution of primes in linear forms.

Abstract

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