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In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf) we find the following result: If the Riemann hypothesis is true ...

\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation} In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....

The Lagarias inequality, which is equivalent to the Riemann hypothesis, is: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n) =:L(n)$$ for all natural numbers $n$, where $\sigma=$ sum of divisors, $H_n=n$-th ...

Robin's inequality $$\sigma_1(n)<e^\gamma n\log\log n$$ at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...