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