Murmurations

Two animations of results from the paper Murmurations: what the phenomenon says, and what the murmuration densities look like far out.

Final frame of the statement animation: two opposite point clouds in teal and purple following the dashed murmuration density.

Theorem 1

Main statement

From the Fourier coefficients of a single modular form to the signed averages over thousands of forms, which trace the murmuration density of Theorem 1.

Still of the weight-42 animation: the teal density nearly following the purple universal limit.

Theorem 3

Universal function for all weights

The weight-42 murmuration density, followed out to √y = 2000, settles onto the universal limit function shared by all weights.

Paper: arXiv:2310.07681