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Linear constellations in primes with arithmetic restrictions

Christopher Frei, Joachim König, Magdaléna Tinková

TL;DR

This work extends Green–Tao type results on linear configurations in primes to primes under arithmetic restrictions, establishing both conditional (HRH-based) and unconditional (Artin-symbol) analogues. The authors develop a comprehensive framework combining nilsequence orthogonality, the W-trick, Hooley’s method, and an ideal von Mangoldt model to handle restricted primes, defining key densities $\delta(a,b,q)$, $\eta_{K,C}$, and $\tau_{\mathbf K,\mathbf C,\Psi}$, with leading terms given by $\mathfrak S$ constants. They prove asymptotics for finite-complexity systems of affine-linear forms and demonstrate an application to inverse Galois theory, producing infinitely many Galois extensions with prescribed local behavior. The results connect analytic number theory with algebraic number theory, providing a robust method to count constellations of primes under prescribed arithmetic constraints and enabling local–global compatibility in Galois realizations. The paper thereby broadens the reach of Green–Tao–Ziegler techniques to structured prime subsets and highlights new avenues for interplay between prime patterns and 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.

Linear constellations in primes with arithmetic restrictions

TL;DR

This work extends Green–Tao type results on linear configurations in primes to primes under arithmetic restrictions, establishing both conditional (HRH-based) and unconditional (Artin-symbol) analogues. The authors develop a comprehensive framework combining nilsequence orthogonality, the W-trick, Hooley’s method, and an ideal von Mangoldt model to handle restricted primes, defining key densities , , and , with leading terms given by constants. They prove asymptotics for finite-complexity systems of affine-linear forms and demonstrate an application to inverse Galois theory, producing infinitely many Galois extensions with prescribed local behavior. The results connect analytic number theory with algebraic number theory, providing a robust method to count constellations of primes under prescribed arithmetic constraints and enabling local–global compatibility in Galois realizations. The paper thereby broadens the reach of Green–Tao–Ziegler techniques to structured prime subsets and highlights new avenues for interplay between prime patterns and 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.
Paper Structure (50 sections, 50 theorems, 364 equations, 3 tables)

This paper contains 50 sections, 50 theorems, 364 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Primes with prescribed primitive roots
  3. Primes with prescribed Artin symbols
  4. Application in inverse Galois theory
  5. Previous results and examples
  6. Notation, outline and proof strategy
  7. Some notation
  8. Local densities
  9. Polynomial subsequences of equidistributed nilsequences
  10. W-trick
  11. Von Mangoldt model and orthogonality with nilsequences
  12. Hooley's method
  13. Completion of proofs
  14. Examples
  15. Local densities
  16. ...and 35 more sections

Key Result

Theorem 1.2

Let $s,t,L,N\in\mathbb{N}$. Let $\mathbf{a}=(a_1,\ldots,a_t)\in\mathbb{Z}^t$ such that no $a_i$ is equal to $-1$ or a perfect square, and assume that $\mathop{\mathrm{HRH}}\nolimits(a_i)$ holds for all $1\leq i\leq t$. Let $\Psi=(\psi_1,\ldots,\psi_t):\mathbb{Z}^s\to\mathbb{Z}^t$ be a system of affi where Moreover, for each prime $p\nmid\mathscr{D}_\mathbf{a}$, we have $\sigma_{\mathbf{a},\Psi}(p

Theorems & Definitions (94)

  • Definition 1.1: Hooley's Riemann hypothesis
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • Example 1.10
  • ...and 84 more