Central limit theorem for Artin $L$-functions

TL;DR

This paper establishes a central limit theorem for Artin and modular form L-functions, demonstrating Gaussian distribution of traces and proving related counting conjectures for specific Galois groups.

Contribution

It proves the central limit theorem for Artin L-functions and verifies counting conjectures for S4 and S5 fields, extending understanding of L-function distributions.

Findings

Sum of Frobenius traces follows Gaussian distribution

Counting conjectures proven for S4 and S5 Galois groups

Central limit theorem shown for modular form L-functions

Abstract

We show that the sum of the traces of Frobenius elements of Artin $L$-functions in a family of $G$-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for $S_4$ and $S_5$-fields. We also show central limit theorem for modular form $L$-functions with the trivial central character with respect to congruence subgroups as the level goes to infinity.