Robin's Criterion on Superabundant Numbers

EasyChair Preprint no. 9250

6 pagesDate: November 5, 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 $n$, $\gamma \approx 0.57721$ 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 $n \mapsto \frac{\sigma(n)}{n}$. Let $P_{n}$ be equal to $\prod_{q \mid \frac{N_{r}}{6}} \frac{q^{\nu_{q}(n) + 2} - 1}{q^{\nu_{q}(n) + 2} - q}$ for a superabundant number $n > 5040$, where $\nu_{p}(n)$ is the $\textit{p-adic}$ order of $n$, $q_{k}$ is the largest prime factor of $n$ and $N_{r} = \prod_{i = 1}^{r} q_{i}$ is the largest primorial number of order $r$ such that $\frac{N_{r}}{6} < q_{k}^{2}$. In this note, we prove that the Riemann hypothesis is true when $P_{n} \geq Q$ holds for all large enough superabundant numbers $n$, where $Q = \frac{1.2 \cdot (2 - \frac{1}{8}) \cdot (3 - \frac{1}{3})}{(2 - \frac{1}{2^{19}}) \cdot (3 - \frac{1}{3^{12}})} \approx 1.0000015809$. In particular, the inequality $P_{n} \geq Q$ holds when $\sum_{q \mid \frac{N_{r}}{6}} \sigma(\frac{n'}{q^{\nu_{q}(n) + 1}}) \geq \sigma(n') \cdot \log Q$ also holds such that $n' = \prod_{q \mid \frac{N_{r}}{6}} q^{\nu_{q}(n) + 1}$ since $\sigma(\ldots)$ is multiplicative.

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