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arXiv · 2307.00256

Murmurations of Dirichlet characters

Kyu-Hwan Lee·Thomas Oliver·Alexey Pozdnyakov
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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.

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Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov. 2023-07-01. Murmurations of Dirichlet characters. https://doi.org/10.1093/imrn%2Frnae277

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