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

T. L. Todorova

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arXiv · 2404.01045

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

T. L. Todorova

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Abstract

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.

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T. L. Todorova. 2024-04-01. On the distribution of $\alpha p^2+\beta$ modulo one for primes $p$ such that $p+2$ has no more two prime divisors. https://arxiv.org/abs/2404.01045

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