A Very Brief Note on the Riemann Hypothesis

EasyChair Preprint no. 8557, version 4

6 pagesDate: August 8, 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 study some properties about the possible counterexamples of the Robin's inequality greater than $5040$. We obtain for the possible smallest counterexample of the Robin's inequality $n > 5040$ that $n < \alpha^{2} \cdot (N_{k})^{1.000208229291}$, where $N_{k} = \prod_{i = 1}^{k} q_{i}$ is the primorial number of order $k$ and $\alpha = \prod_{i=1}^{k} \left(1 - \frac{1}{q_{i}^{a_{i} + 1}} \right)$ when $n$ is a superabundant number by this representation $n = \prod_{i = 1}^{k} q_{i}^{a_{i}}$. Finally, we provide solid arguments which suggest that the Riemann Hypothesis is possibly true.

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