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\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)....

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 ...