On the Least Colossally Abundant Exception to Robin's Inequality

Abstract

Robin's Inequality posits $G(n)<e^{\gamma}$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^{\gamma}\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By analyzing the prime or semiprime quotient $\displaystyle\frac{n}{m}$ for consecutive Colossally Abundant (CA) numbers $m$ followed by $n$ (where $m$ satisfies Robin's Inequality and $n$ violates it), we demonstrate that if the Riemann Hypothesis is false, then the least CA counterexample, $n$, must be constrained to the band $e^\gamma<G(n)<e^\gamma \left(1+\displaystyle\frac{c}{(\log n)^b}\right)$ where $0 < b < 1/2$, i.e. excluded from the infinite set beyond the higher threshold.