I have requested a pdf of Robin 1984 from campus scanning service. One highlight of the article that really should be mentioned is this:
For $n \geq 13,$ we have $$ \sigma(n) \; < \; \; e^\gamma \; n \log \log n \; + \; \frac{ \; 0.64821364942... \; \; n \; }{\log \log n},$$ with the constant in the numerator giving equality for $n=12.$
see:
Which $n$ maximize $G(n)=\frac{\sigma(n)}{n \log \log n}$?
That, at least, rests on effective bounds of Rosser and Schoenfeld (1962), which can be downloaded from ROSSER
Well, maybe not so directly. R+S do the unconditional bound for $n/\phi(n)$ in Theorem 15, pages 71-72, formulas (3.41) and (3.42). The treatment for $\sigma(n)$ is quite similar in spirit, maybe Robin was the first to write it down. The analogue of the primorials PRIMO and $n^{1-\delta}/\phi(n)$ is the colossally abundant CA numbers and $\sigma(n)/ n^{1 + \delta}.$