On Robin's criterion for RH

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.

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

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...

From Alaoglu and Erdos, the maximum value, for some $0 < \delta \leq 1, $ of $$ \frac{\sigma(n)}{ n^{1+\delta}} $$ occurs when the prime factor $p$ of $n$ has exponent $$ v_p(n) = \left\lfloor \frac{\log (p^{1 + \delta} - 1) - \log(p^\delta - 1)}{\log p} \right\rfloor \; - \; 1. $$
This is Theorem 10 on page 455. The results of this construction are the colossally abundant numbers. The construction is originally due to Ramanujan, but the part of his manuscript that dealt with ca numbers was not printed owing to paper shortages at the time.

Hardy and Wright use $d(n)$ for the number of divisors of $n.$ This is in the original paper by Ramanujan. For some $0 < \delta \leq 1, $ the maximum of $$ \frac{d(n)}{ n^{\delta}} $$ occurs when the prime factor $p$ of $n$ has exponent $$ v_p(n) = \left\lfloor \frac{1}{p^\delta - 1} \right\rfloor. $$ The results are called the superior highly composite numbers SHC.

So, taking all three with $\delta = 1/2,$ we get lemmas $$ \phi(n) \geq \sqrt{\frac{n}{2}}, \; \; d(n) \leq \sqrt{3n}, \; \; \sigma(n) \leq 3 \left( \frac{n}{2} \right)^{3/2}. $$

In all three cases, if $\delta$ is such that more than one number $n$ achieves the maximum value of the ratio specified, we are choosing the largest of these $n$'s.

I cannot find Robin's paper either (thank you, Elsevier), but a stronger theorem was proved by Jeff Lagarias in 2002:

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Amer. Math. Monthly 109 (2002), 534–543.

Lagarias' statement is:

The RH is true if and only if $ \sigma(n) < H_n + \exp(H_n) \log(H_n), $ where $H_n$ are the usual harmonic numbers.