A Very Brief Note on the Riemann Hypothesis

EasyChair Preprint 8557, version 9

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

@booklet{EasyChair:8557,
  author    = {Frank Vega},
  title     = {A Very Brief Note on the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 8557},
  year      = {EasyChair, 2022}}