In G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213

A results is:

If the Riemann hypothesis is True and $n ≥ 5041$
$\frac{\sigma(n)}{n} < e^\gamma \ln \ln (n)$

We also know that $e^\gamma < e$

Without using the Riemann hypothesis, it is possible to show that:

$\frac{\sigma(n)}{n} < e \ln \ln (n)$