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
\[\sigma(n) < e^{\gamma} n \log \log n\]$$
\sigma(n) < e^{\gamma} n \log \log n
$$
for sufficiently large $n$ is not due to Robin, but due to Ramanujan.
And still before that Grönwald (1913) showed (uncoditionally)
\[\limsup_{n\to \infty}\frac{\sigma(n)}{n \log \log n} = e^{\gamma}\]$$
\limsup_{n\to \infty}\frac{\sigma(n)}{n \log \log n} = e^{\gamma}
$$
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
\[ \sum_{n=1}^{\infty} \frac{\sigma_y(n)}{n^s} = \zeta(s)\zeta(s-y)\]$$
\sum_{n=1}^{\infty} \frac{\sigma_y(n)}{n^s} = \zeta(s)\zeta(s-y)
$$
so
\[ \sum_{n=1}^{\infty} \frac{\sigma (n)}{n^s} = \zeta(s)\zeta(s-1).\]$$
\sum_{n=1}^{\infty} \frac{\sigma (n)}{n^s} = \zeta(s)\zeta(s-1).
$$
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.