Abstract
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 \(\sup \{\Re (\rho )~|~\zeta (\rho ) = 0\}\) is not equal to 1.
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Acknowledgements
We are grateful to Jean-Louis Nicolas for his encouragement and helpful correspondence. We are also grateful to the referees for their feedback on earlier drafts of this paper.
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Cavendish, W., Tsimerman, J. Towards an elementary formulation of the Riemann hypothesis in terms of permutation groups. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00829-2
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DOI: https://doi.org/10.1007/s11139-024-00829-2