Linear constellations in primes with arithmetic restrictions

TL;DR

This paper extends Green and Tao's theorem on linear constellations in primes to primes with specific arithmetic restrictions, providing both conditional and unconditional results with applications in inverse Galois theory.

Contribution

It introduces new theorems for primes with prescribed primitive roots or Artin symbols, advancing understanding of primes under arithmetic constraints.

Findings

Conditional proof assuming Hooley's Riemann hypothesis

Unconditional proof with Artin symbol restrictions

Application in inverse Galois theory

Abstract

We prove analogues of the theorem of Green and Tao on linear constellations in primes, in which the primes under consideration are restricted by certain arithmetic conditions. Our first main result is conditional upon Hooley's Riemann hypothesis and imposes the extra condition that the primes have prescribed primitive roots. Our second main result is unconditional and imposes the extra condition that the primes have prescribed Artin symbols in given Galois number fields. In the appendix we present an application of the second result in inverse Galois theory.