On the Least Colossally Abundant Exception to Robin's Inequality

TL;DR

This paper investigates the least counterexample to Robin's Inequality assuming the Riemann Hypothesis is false, showing it must lie within a specific narrow band near the critical value.

Contribution

It introduces a novel analysis of prime or semiprime quotients between consecutive Colossally Abundant numbers to constrain potential counterexamples to Robin's Inequality.

Findings

If RH is false, the least CA counterexample is confined to a specific logarithmic band.

The analysis excludes the existence of counterexamples beyond a certain threshold.

Provides a new perspective on the distribution of CA numbers relative 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.