A Very Brief Note on the Riemann Hypothesis

EasyChair Preprint no. 8557, version 17

6 pagesDate: September 21, 2022

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 $n$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. In this note, using Robin's inequality on superabundant numbers, we prove that the Riemann Hypothesis is true. This proof is an extension of the article Robin's criterion on divisibility'' published by The Ramanujan Journal on May 3rd, 2022.

Keyphrases: prime numbers, Riemann hypothesis, Robin's inequality, sum-of-divisors function, Superabundant numbers