# Primes with Beatty and Chebotarev conditions

## TL;DR

This paper investigates primes in Beatty sequences with algebraic splitting conditions, establishing their density, distribution, and bounded gaps, and generalizing key results like Green--Tao theorem.

## Contribution

It provides a new density formula for primes in Beatty sequences with Chebotarev conditions and proves bounded gaps and a generalized Green--Tao theorem for such primes.

## Key findings

- Density of primes in Beatty and Chebotarev sets equals the product of individual densities.
- Primes in the intersection satisfy a Bombieri--Vinogradov type theorem.
- Existence of bounded gaps for primes in these specialized sets.

## Abstract

We study the prime numbers that lie in Beatty sequences of the form $\lfloor \alpha n + \beta \rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $\alpha^{-1}\cdot\frac{|C|}{|G|}$. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri--Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bounded gaps result and the Green--Tao theorem.