Moments in the Chebotarev density theorem: general class functions

Régis de La Bretèche (1) , Daniel Fiorilli (2) , Florent Jouve (3)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 LMO - Laboratoire de Mathématiques d'Orsay
3 IMB - Institut de Mathématiques de Bordeaux

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\"{\i}che, 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.

Mots clés: Chebotarev density theorem; Moment computations; Explicit formulae in arithmetic

Domaines: Théorie des nombres [math.NT]

Régis de La Bretèche, Daniel Fiorilli, Florent Jouve. Moments in the Chebotarev density theorem: general class functions. 2023. ⟨hal-03961980⟩