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 $e^\gamma < e$ , Now my question here is :

>>**Question:** Without using the Riemann hypothesis, is it possible to show that:
$\frac{\sigma(n)}{n} <  e \ln \ln (n)$ , $n\geq 5041$?