Central limit theorem for Artin L-functions

Abstract

We show that the sum of the traces of Frobenius elements of Artin [Formula: see text]-functions in a family of [Formula: see text]-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for [Formula: see text] and [Formula: see text]-fields. We also prove a central limit theorem for the [Formula: see text]-functions of modular forms on congruence subgroups [Formula: see text] as [Formula: see text].