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 $G(n)>e^{\gamma }\left(1+\frac{c}{(\log n{)}^b}\right)$ for infinitely many values of $n$. By analyzing the prime or semiprime quotient $\frac{n}{m}$ 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 $e^{\gamma }<G(n)<e^{\gamma }\left(1+\frac{c}{(\log n{)}^b}\right)$ where $0<b<1/2$, i.e. excluded from the infinite set beyond the higher threshold.