Murmurations of Dirichlet characters

TL;DR

This paper introduces new density calculations for Dirichlet characters, revealing universal properties and phase transition behaviors that connect to broader themes in number theory and L-functions.

Contribution

It computes murmuration densities for two Dirichlet character families, demonstrating universality and phase transition phenomena in their distributions.

Findings

First density aligns with experimental murmuration patterns.

Second density shows universality similar to holomorphic newforms.

Density interpolates phase transition in 1-level density for symplectic L-functions.

Abstract

We calculate murmuration densities for two families of Dirichlet characters. The first family contains complex Dirichlet characters normalized by their Gauss sums. Integrating the first density over a geometric interval yields a murmuration function compatible with experimental observations. The second family contains real Dirichlet characters weighted by a smooth function with compact support. We show that the second density exhibits a universality property analogous to Zubrilina's density for holomorphic newforms, and it interpolates the phase transition in the the $1$-level density for a symplectic family of $L$-functions.