Murmurations and ratios conjectures
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909587206045696 |
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| author | Cowan, Alex |
| author_facet | Cowan, Alex |
| contents | 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 $ζ'/ζ$ 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 $ζ$ on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12723 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Murmurations and ratios conjectures Cowan, Alex Number Theory 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 $ζ'/ζ$ 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 $ζ$ on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures. |
| title | Murmurations and ratios conjectures |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.12723 |