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arXiv · 2306.10425

Murmurations and explicit formulas

Alex Cowan
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Abstract

Unexpected oscillations in $a_p$ values in a family of elliptic curves were observed experimentally by He, Lee, Oliver, and Pozdnyakov. We propose a heuristic explanation for these oscillations based on the "explicit formula" from analytic number theory. A crucial ingredient in this heuristic is that the distribution of the zeros of the associated $L$-functions has a quasi-periodic structure. We present empirical results for a family of elliptic curves, a family of quadratic Dirichlet characters whose values exhibit similar oscillations, and a family of Dirichlet characters whose values do not.

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Alex Cowan. 2023-06-17. Murmurations and explicit formulas. https://arxiv.org/abs/2306.10425

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