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Moments in the Chebotarev density theorem: general class functions

Régis de La Bretèche, Daniel Fiorilli, Florent Jouve

TL;DR

The paper develops a framework to bound lower moments of the Chebotarev error term in Galois extensions by using a weighted explicit formula and positivity, avoiding linear independence hypotheses for zeros. It introduces a general weighted moment $\widetilde{M}_{2m}(U,L/K;t,\eta,\Phi)$ that incorporates a class function $t$, smooth weight $\eta$, and a test $\Phi$, and shows these moments are at least Gaussian with variance $\nu(L/F,t^+;\eta)$ under GRH and Artin's conjecture; the variance is expressed via zeros of Artin $L$-functions and ramification data, with the size controlled by norms $\lambda_{j,k}(t)$ and the obstruction $S_t$. The authors provide explicit applications to families such as dihedral, radical, irreducible-character-oriented, and symmetric-group ($S_n$) extensions, giving concrete lower bounds and explaining how the arithmetic invariants (root discriminant, Artin conductors) influence the bounds. Overall, the work yields explicit, uniform lower bounds for moments of the Chebotarev error term in broad non-abelian settings and demonstrates how positivity and zero-sum analyses can yield Gaussian-type oscillations in Chebotarev counts, with potential consequences for Ω-type results in counting primes in arithmetic settings.

Abstract

In this paper we find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bella\''ıche, we consider general class functions and prove bounds which depend on norms associated to these functions. Our bounds also involve the ramification and Galois theoretical information of the underlying extension $L/K$. Under a natural condition on class functions (which appeared in earlier work), we obtain that those moments are at least Gaussian. The key tools in our approach are the application of positivity in the explicit formula followed by combinatorics on zeros of Artin $L$-functions (which generalize previous work), as well as precise bounds on Artin conductors.

Moments in the Chebotarev density theorem: general class functions

TL;DR

The paper develops a framework to bound lower moments of the Chebotarev error term in Galois extensions by using a weighted explicit formula and positivity, avoiding linear independence hypotheses for zeros. It introduces a general weighted moment that incorporates a class function , smooth weight , and a test , and shows these moments are at least Gaussian with variance under GRH and Artin's conjecture; the variance is expressed via zeros of Artin -functions and ramification data, with the size controlled by norms and the obstruction . The authors provide explicit applications to families such as dihedral, radical, irreducible-character-oriented, and symmetric-group () extensions, giving concrete lower bounds and explaining how the arithmetic invariants (root discriminant, Artin conductors) influence the bounds. Overall, the work yields explicit, uniform lower bounds for moments of the Chebotarev error term in broad non-abelian settings and demonstrates how positivity and zero-sum analyses can yield Gaussian-type oscillations in Chebotarev counts, with potential consequences for Ω-type results in counting primes in arithmetic settings.

Abstract

In this paper we find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bella\''ıche, we consider general class functions and prove bounds which depend on norms associated to these functions. Our bounds also involve the ramification and Galois theoretical information of the underlying extension . Under a natural condition on class functions (which appeared in earlier work), we obtain that those moments are at least Gaussian. The key tools in our approach are the application of positivity in the explicit formula followed by combinatorics on zeros of Artin -functions (which generalize previous work), as well as precise bounds on Artin conductors.
Paper Structure (14 sections, 25 theorems, 207 equations)

This paper contains 14 sections, 25 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Explicit Families of Galois extensions and class functions
  3. Dihedral extensions
  4. Radical extensions
  5. Moments for irreducible characters
  6. $S_n$-extensions
  7. Artin conductors
  8. Sums over zeros of Artin $L$-functions
  9. Proof of Theorems \ref{['theorem main']} and \ref{['theorem main 2']}: Induction and positivity
  10. Application to specific extensions and class functions: proofs
  11. $D_n$-examples
  12. Example of a radical extension
  13. Real parts of characters as class functions
  14. $S_n$-extensions

Key Result

Theorem 1.1

Let $L/K/F$ be a tower of number fields such that $L\neq \mathbb Q$, $L/F$ is Galois, and assume GRH and AC for the extension Note that AC for the extension $L/F$ implies AC for the extension $L/K$.$L/F$. Define $G:={\rm Gal}(L/K)$, $G^+:={\rm Gal}(L/F)$, let $\eta \in \mathcal{S}_\delta$, $\Phi \in where $\kappa_\eta>0$ is a constant which depends only on $\eta$ and

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Example
  • Remark 1.8
  • Corollary 1.9
  • ...and 46 more