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

TL;DR

This paper explores a permutation group-based inequality related to the Riemann hypothesis, showing it holds if the hypothesis is true and providing conditions for its potential converse.

Contribution

It introduces a new formulation of the Riemann hypothesis using permutation groups and advances understanding of its equivalence through group-theoretic inequalities.

Findings

Proves the inequality g(n) ≤ e^{√p_n} under the Riemann hypothesis

Shows that if the Riemann hypothesis is false, the inequality can be violated for some n

Links the inequality to the distribution of zeros of the Riemann zeta function

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.