The First Hypothetical Counterexample of Robin's Criterion

Robin's criterion states that the Riemann Hypothesis is true if and only if the inequality \( \sigma(n) < e^{\gamma} \cdot n \cdot \log \log n \) holds for all \( n > 5040 \), where \( \sigma(n) \) is the sum-of-divisors function of \( n \)and \( \gamma \approx 0.57721 \) is the Euler-Mascheroni constant. We require the properties of superabundant numbers, that is to say left to right maxima of \( n \mapsto \frac{\sigma(n)}{n} \). Akbary and Friggstad showed that the least counterexample to Robin's criterion, should one exist, must be a superabundant number. Using this result, we prove that the Robin's inequality cannot fail for such hypothetical counterexample, which yields a proof of the Riemann Hypothesis. This work refines the approach taken in the author's earlier article ``Robin's criterion on divisibility'', published in The Ramanujan Journal.