Download PDFOpen PDF in browserWhen the Riemann Hypothesis Might Be FalseEasyChair Preprint 68447 pages•Date: October 15, 2021AbstractRobin 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 sumofdivisors function and $\gamma \approx 0.57721$ is the EulerMascheroni 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 HardyRamanujan integer. If the Riemann Hypothesis is false, then there are infinitely many HardyRamanujan 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 $m$. Keyphrases: Riemann hypothesis, Robin inequality, prime numbers, sumofdivisors function
