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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2108.09570 |
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Table of 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.