On the distribution of $αp^2+β$ modulo one for primes $p$ such that $p+2$ has no more two prime divisors

T. L. Todorova

Published 2024-04-01, updated 2024-04-04Version 2

A classical problem in analytic number theory is to study the distribution of fractional part $\alpha p^k+\beta,\,k\ge 1$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. For $k=2$ 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.