Abstract
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 natural numbers \(n > 5040\), where \(\sigma(n)\) is the sum-of-divisors function of \(n\), \(\gamma \approx 0.57721\) is the Euler-Mascheroni constant and \(\log\) is the natural logarithm. We require the properties of superabundant numbers, that is to say left to right maxima of \(n \mapsto \frac{\sigma(n)}{n}\). Let \(P_{n}\) be equal to \(\prod_{q \mid \frac{N_{r}}{6}} \frac{q^{\nu_{q}(n) + 2} - 1}{q^{\nu_{q}(n) + 2} - q}\) for a superabundant number \(n > 5040\), where \(\nu_{p}(n)\) is the \(\textit{p-adic}\) order of \(n\), \(q_{k}\) is the largest prime factor of \(n\) and \(N_{r} = \prod_{i = 1}^{r} q_{i}\) is the largest primorial number of order \(r\) such that \(\frac{N_{r}}{6} < q_{k}^{2}\). In this note, we prove that the Riemann hypothesis is true when \(P_{n} \geq Q\) holds for all large enough superabundant numbers \(n\), where \(Q = \frac{1.2 \cdot (2 - \frac{1}{8}) \cdot (3 - \frac{1}{3})}{(2 - \frac{1}{2^{19}}) \cdot (3 - \frac{1}{3^{12}})} \approx 1.0000015809\). In particular, the inequality \(P_{n} \geq Q\) holds when \(\sum_{q \mid m} \sigma(\frac{m}{q^{\nu_{q}(n) + 1}}) \gtrapprox \sigma(m) \cdot \log Q\) also holds such that \(m = \prod_{q \mid N_{r}} q^{\nu_{q}(n) + 1}\) since \(\sigma(\ldots)\) is multiplicative.