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arXiv · 2408.12723

Murmurations and ratios conjectures

Alex Cowan
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Abstract

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a fixed elliptic curve, and the inverse Mellin transform of the shifted second moment of $\zeta'/\zeta$ on vertical lines; assuming GRH, for primitive quadratic Dirichlet characters, and holomorphic modular forms of prime level tending to infinity with sign and weight fixed; and unconditionally the inverse Mellin transform of the shifted second moment of $\zeta$ on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures.

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Alex Cowan. 2024-08-22. Murmurations and ratios conjectures. https://arxiv.org/abs/2408.12723

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