In view of OP's comment on Igor Rivin's answer it seems that the 'actual' question could be something else.

The inequality under RH
$$
\sigma(n) < e^{\gamma} n \log \log n
$$ 
for sufficiently large $n$ is not due to Robin, but due to Ramanujan.
And still before that Grönwald (1913) showed (uncoditionally)
$$
\limsup_{n\to \infty}\frac{\sigma(n)}{n \log \log n} = e^{\gamma}
$$

As to why questions like this are linked to RH at all.
For example, recall that if one defines $\sigma_y (n) = \sum_{d|n}d^y$ then 
for the asociated Dirichlet series one has 
$$
\sum_{n=1}^{\infty} \frac{\sigma_y(n)}{n^s} = \zeta(s)\zeta(s-y)
$$
so  
$$
\sum_{n=1}^{\infty} \frac{\sigma (n)}{n^s} = \zeta(s)\zeta(s-1).
$$
 
Without having followed up on the precise historical deveopment it seems rather like so:
one studies the growth of $\sigma$ as for plenty of other arithmetical functions. 
Somebody (Grönwald) shows a nice result, somebody else (Ramanujan) shows something more precise *under* RH. Then somebody (Robin) decdides to investigate whether this is in fact *equivalent* to RH (as some other results known under RH, most notably the asymptotic count of prime numbers).

This seems like a quite natural development to me.