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 and $n ≥ 5041$, $\frac{\sigma(n)}{n} < e^\gamma \ln \ln (n)$

We also know that $e^\gamma < e$. Now my question here is:

Question: Without using the Riemann hypothesis, is it possible to show that $\frac{\sigma(n)}{n} < e \ln \ln (n)$ ?