When the Riemann Hypothesis Might Be False

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality \(\sigma(n) < e^{\gamma } \times n \times \log \log n\) holds for all natural numbers \(n > 5040\), where \(\sigma(n)\) is the sum-of-divisors function and \(\gamma \approx 0.57721\) is the Euler-Mascheroni constant. Let \(q_{1} = 2, q_{2} = 3, \ldots, q_{m}\) denote the first \(m\) consecutive primes, then an integer of the form \(\prod_{i=1}^{m} q_{i}^{a_{i}}\) with \(a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0\) is called an Hardy-Ramanujan integer. If the Riemann Hypothesis is false, then there are infinitely many Hardy-Ramanujan integers \(n > 5040\) such that Robin inequality does not hold and \(n < (4.48311)^{m} \times N_{m}\), where \(N_{m} = \prod_{i = 1}^{m} q_{i}\) is the primorial number of order \(m\).