Robin's Criterion on Divisibility

EasyChair Preprint no. 7708

11 pagesDate: April 2, 2022

Abstract

Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ 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 show that the Robin inequality is true for all natural numbers $n > 5040$ that are not divisible by some prime between $2$ and $1771559$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6} \times \log\log n' \leq \log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n > 5040$ of the Robin inequality implies that $q_{m} > e^{31.018189471}$, $1 < \frac{(1 + \frac{1.2762}{\log q_{m}}) \times \log(2.82915040011)}{\log \log n}+ \frac{\log N_{m}}{\log n}$, $(\log n)^{\beta} < 1.03352795481\times\log(N_{m})$ and $n < (2.82915040011)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$, $q_{m}$ is the largest prime divisor of $n$ and $\beta = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$.

Keyphrases: prime numbers, Riemann hypothesis, Riemann zeta function, Robin inequality, sum-of-divisors function