Towards an elementary formulation of the Riemann hypothesis in terms of permutation groups

This paper investigates the relationship between the Riemann hypothesis and the statement \(\forall n, ~g(n) \le e^{\sqrt{p_n}}\) $\forall n,g(n)\le e^{\sqrt{p_n}}$ , where g(n) is the maximum order of an element of \(S_n\) $S_n$ , the symmetric group on n elements, and \(p_n\) $p_n$ is the n-th prime. We show this inequality holds under the Riemann Hypothesis. We also make progress towards establishing the converse by proving \(\exists n,~g(n)>e^{\sqrt{p_n}}\) $\exists n,g(n)>e^{\sqrt{p_n}}$ if the Riemann Hypothesis is false and the supremum of the set of the real parts of the Riemann zeta function’s zeros \(\sup \{\Re (\rho )~|~\zeta (\rho ) = 0\}\) $sup\{\Re (\rho )|\zeta (\rho )=0\}$ is not equal to 1.