Robin's Criterion on Superabundant Numbers

EasyChair Preprint no. 8992, version 3

Versions: 123history
6 pagesDate: October 11, 2022

Abstract

A 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 non-trivial 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 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 $n \mapsto \frac{\sigma(n)}{n}$. 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: prime numbers, Riemann hypothesis, Robin's inequality, sum-of-divisors function, Superabundant numbers