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 $\sigma(n)/ n^{1 + \delta}.$


Well, I am not sure where it is written down, but it is easy enough to show that the maximum value, for some $0 < \delta \leq 1, $ of
$$  \frac{ n^{1-\delta}}{\phi(n)}  $$
occurs when the prime factor $p$ of $n$ has exponent
$$     v_p(n) = \left \lfloor  \frac{p^{1-\delta}}{p-1}  \right \rfloor.$$
Since, for a fixed $\delta,$ this expression is either 0 or 1 and nonincreasing in $p,$ it turns out that the optima occur at the primorials, the products of the consecutive primes from 2 to something...  


  [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