Moments in the Chebotarev density theorem: general class functions

TL;DR

This paper establishes lower bounds on higher moments of the error term in the Chebotarev density theorem for general class functions, revealing Gaussian behavior under certain conditions.

Contribution

It extends previous work by deriving bounds for moments involving general class functions and linking them to Galois and ramification data.

Findings

Moments are at least Gaussian under certain conditions.

Bounds depend on norms of class functions and Galois data.

Application of positivity and zero combinatorics of Artin L-functions.

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\''{\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.