Integer moments of the derivatives of the Riemann zeta function
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912578803859456 |
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| author | Hughes, Christopher Pearce-Crump, Andrew |
| author_facet | Hughes, Christopher Pearce-Crump, Andrew |
| contents | We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a conjecture for the integer moments of mixed derivatives of the zeta function. This generalises a result of the authors where they took complex moments of the first derivative of the zeta function, evaluated at the non-trivial zeros. We approach this problem in two different ways: the first uses a random matrix theory approach, and the second by the Ratios Conjecture of Conrey, Farmer, and Zirnbauer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07792 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integer moments of the derivatives of the Riemann zeta function Hughes, Christopher Pearce-Crump, Andrew Number Theory 11M06, 11M26, 11M50 We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a conjecture for the integer moments of mixed derivatives of the zeta function. This generalises a result of the authors where they took complex moments of the first derivative of the zeta function, evaluated at the non-trivial zeros. We approach this problem in two different ways: the first uses a random matrix theory approach, and the second by the Ratios Conjecture of Conrey, Farmer, and Zirnbauer. |
| title | Integer moments of the derivatives of the Riemann zeta function |
| topic | Number Theory 11M06, 11M26, 11M50 |
| url | https://arxiv.org/abs/2509.07792 |