Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866917756785393664 |
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| author | Cavendish, Will Tsimerman, Jacob |
| author_facet | Cavendish, Will Tsimerman, Jacob |
| contents | This paper investigates the relationship between the Riemann hypothesis and the statement $\forall n, ~g(n) \le e^{\sqrt{p_n}}$, where $g(n)$ is the maximum order of an element of $S_n$, the symmetric group on $n$ elements, and $p_n$ is the $n$-th prime. We show this inequality holds under the Riemann Hypothesis. We also make progress towards establishing the converse by proving $\exists n,~g(n)>e^{\sqrt{p_n}}$ if the Riemann Hypothesis is false and the supremum of the set of the real parts of the Riemann zeta function's zeros $\sup \{\Re(ρ)~|~ζ(ρ) = 0\}$ is not equal to 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_09570 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups Cavendish, Will Tsimerman, Jacob Number Theory This paper investigates the relationship between the Riemann hypothesis and the statement $\forall n, ~g(n) \le e^{\sqrt{p_n}}$, where $g(n)$ is the maximum order of an element of $S_n$, the symmetric group on $n$ elements, and $p_n$ is the $n$-th prime. We show this inequality holds under the Riemann Hypothesis. We also make progress towards establishing the converse by proving $\exists n,~g(n)>e^{\sqrt{p_n}}$ if the Riemann Hypothesis is false and the supremum of the set of the real parts of the Riemann zeta function's zeros $\sup \{\Re(ρ)~|~ζ(ρ) = 0\}$ is not equal to 1. |
| title | Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2108.09570 |