Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups

2021/08/21 by Will Cavendish, Jacob Tsimerman, Cavendish, Will +1 · 1 citation
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2108.09570

openalex publication_date 2021/08/21 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

This paper investigates the relationship between the Riemann hypothesis and the statement ∀ n, ~g(n) ≤ e^√(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 ∃ n,~g(n)>e^√(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.

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