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Robin’s Criterion on Superabundant Numbers
Version 1
: Received: 4 August 2024 / Approved: 5 August 2024 / Online: 6 August 2024 (06:56:30 CEST)
How to cite: Vega, F. Robin’s Criterion on Superabundant Numbers. Preprints 2024, 2024080348. https://doi.org/10.20944/preprints202408.0348.v1 Vega, F. Robin’s Criterion on Superabundant Numbers. Preprints 2024, 2024080348. https://doi.org/10.20944/preprints202408.0348.v1
Abstract
The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. 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 $n \mapsto \frac{\sigma(n)}{n}$. In this note, using Robin's criterion on superabundant numbers, we prove that the Riemann hypothesis is true.
Keywords
Riemann hypothesis; Robin's criterion; superabundant numbers; prime numbers
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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