Murmurations and ratios conjectures

TL;DR

This paper introduces a novel method based on random matrix theory to study murmurations and related phenomena in number theory, providing new insights under various conjectural and unconditional assumptions.

Contribution

It presents a new approach to analyze murmurations using ratios conjectures and the approximate functional equation, applicable to elliptic curves, zeta functions, and modular forms.

Findings

Exhibits murmurations assuming ratios conjectures for elliptic curves and zeta functions.

Demonstrates murmurations under GRH for quadratic Dirichlet characters and modular forms.

Provides an unconditional analysis of the inverse Mellin transform of the zeta function.

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.