Properties of Prime Products and Robin's Inequality

Abstract

There are many formulations of problems that have been proven to be equivalent to the Riemann Hypothesis in modern mathematics. In this paper we look at the formulation of an inequality derived by Robin in 1984 that proves the Riemann Hypothesis true if and only if Robin's Inequality is true for all $n\geq 5040$. In this paper we look at explicit products of prime numbers and show that if a given $n=p_1^{k_1}p_2^{k_2}\ldots p_m^{k_m}$ satisfies Robin's Inequality, then $n=P_1^{k_1}P_2^{k_2}\ldots P_m^{k_m}$ with $p_j \leq P_j$ also satisfies the inequality. We also then offer two conjectures that if proven individually could imply some interesting properties of the Riemann Hypothesis and Robin's Inequality.