Moments in the Chebotarev density theorem: general class functions

de la Bretèche, Régis; Fiorilli, Daniel; Jouve, Florent

doi:10.2140/ant.2025.19.481

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Moments in the Chebotarev density theorem: general class functions

Régis de la Bretèche, Daniel Fiorilli and Florent Jouve

Vol. 19 (2025), No. 3, 481–520

DOI: 10.2140/ant.2025.19.481

Abstract

We find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bellaïche, 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.

Keywords

Chebotarev density theorem, distribution of primes, moments computation

Mathematical Subject Classification

Primary: 11R42, 11R44, 11R45

Secondary: 11N64

Milestones

Received: 6 February 2023

Revised: 19 December 2023

Accepted: 23 May 2024

Published: 20 February 2025

Authors

Régis de la Bretèche
Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Cité, Sorbonne Université CNRS UMR 7586, Paris France

Daniel Fiorilli
Institut de mathématiques d’Orsay Université Paris Saclay Orsay France

Florent Jouve
Université de Bordeaux CNRS UMR 5251 Bordeaux INP Talence France

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Vol. 19, No. 3, 2025