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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.14253 |
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| _version_ | 1866917737707601920 |
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| author | Hughes, Christopher Martin, Greg Pearce-Crump, Andrew |
| author_facet | Hughes, Christopher Martin, Greg Pearce-Crump, Andrew |
| contents | Shanks conjectured that $ζ' (ρ)$, where $ρ$ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, including a generalisation to all higher-order derivatives $ζ^{(n)}(s)$, for which the sign of the mean alternatives between positive for odd $n$ and negative for even $n$. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of $ζ^{(n)}(ρ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_14253 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A heuristic for discrete mean values of the derivative of the Riemann zeta function Hughes, Christopher Martin, Greg Pearce-Crump, Andrew Number Theory Shanks conjectured that $ζ' (ρ)$, where $ρ$ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, including a generalisation to all higher-order derivatives $ζ^{(n)}(s)$, for which the sign of the mean alternatives between positive for odd $n$ and negative for even $n$. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of $ζ^{(n)}(ρ)$. |
| title | A heuristic for discrete mean values of the derivative of the Riemann zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2305.14253 |