The Complete Proof of the Riemann Hypothesis

EasyChair Preprint no. 6710, version 3

Versions: 123history
3 pagesDate: October 4, 2021

Abstract

Robin 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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

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