Download PDFOpen PDF in browserRobin's Criterion on Superabundant NumbersEasyChair Preprint 8992, version 36 pages•Date: October 11, 2022AbstractA trustworthy proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics by several authors. The Riemann hypothesis is the assertion that all nontrivial zeros of the Riemann zeta function have real part $\frac{1}{2}$. 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 sumofdivisors function of $n$, $\gamma \approx 0.57721$ is the EulerMascheroni constant and $\log$ is the natural logarithm. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. If the Riemann hypothesis is false, then there are infinitely many superabundant numbers $n$ such that the Robin's inequality is unsatisfied. In this note, we show that the Robin's inequality always holds for large enough superabundant numbers. By reductio ad absurdum, we prove that the Riemann hypothesis is true. Keyphrases: Riemann hypothesis, Robin's inequality, Superabundant numbers, prime numbers, sumofdivisors function
