On the distribution of \alpha p modulo one for primes p of a special form

TL;DR

This paper investigates the distribution of irrational multiples of primes, specifically those primes where p+2 is almost-prime, showing that their distribution modulo 1 shares similar properties with the general case.

Contribution

It extends the understanding of prime distributions by analyzing a special subsequence where p+2 is almost-prime, revealing similar distribution properties.

Findings

Distribution of lpha p for primes with p+2 almost-prime is similar to general primes

Supports the conjecture about the distribution of primes in special forms

Advances knowledge on primes related to almost-primes

Abstract

A classical problem in analytic number theory is to study the distribution of $\alpha p$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is an almost-prime (the existence of infinitely many such $p$ is another topical result in prime number theory) and prove that its distribution has a similar property.