{"paper":{"title":"Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jacob Tsimerman, Will Cavendish","submitted_at":"2021-08-21T19:39:40Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.09570","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.09570/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}