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:  
http://mathoverflow.net/questions/79927/which-n-maximize-gn-frac-sigmann-log-log-n/79987#79987 

That, at least, rests on effective bounds of Rosser and Schoenfeld (1962), which can be downloaded from [ROSSER][1] 

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][2] and $n^{1-\delta}/\phi(n)$ is the colossally abundant [CA][3] numbers and $\sigma(n)/ n^{1 + \delta}.$


  [1]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255631807
  [2]: http://oeis.org/A002110
  [3]: http://oeis.org/A004490