Alex Cowan
Published 2023-06-17Version 1
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 both a family of elliptic curves as well as a family of quadratic Dirichlet characters whose values exhibit similar oscillations.
Analytic problems for elliptic curves
Computational methods and experiments in analytic number theory
On the nature of the "explicit formulas" in analytic number theory - a simple example
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