EEDDIITT: this gives a proof of my main claim in my first answer, that a certain function takes its minimum value at a certain primorial. I actually put that information, with a few examples, into the wikipedia article, but it was edited out within a minute as irrelevant. No accounting for taste.

ORIGINAL: We take as given Theorem 327 on page 267 of Hardy and Wright, that for some fixed $0 < \delta < 1,$ the function
$$ g(n) = \frac{\phi(n)}{n^{1-\delta}} $$
goes to infinity as $n$ goes to infinity.

Note that $g(1) = 1$ but $g(2) < 1.$ For some $N_\delta,$ whenever $ n > N_\delta$ we get $g(n) > 1.$ It follows that, checking all $1 \leq n \leq N_\delta,$ the quantity $g(n)$ assumes a minimum which is less than 1. Perhaps it assumes this minimum at more than one point. If so, we are taking the largest such value of $n.$

Here we are going to prove that the value of $n$ at which the minimum occurs is the primorial created by taking the product of all the primes $p$ that satisfy
$$ p^{1-\delta} \geq p-1. $$
As I mentioned, in case two the minimum occurs at two different $n,$ this gives the larger of the two.

So, the major task of existence is done by Hardy and Wright. We have the minimum of $g$ at some
$$ n = p_1^{a_1} p_2^{a_2} p_3^{a_3} \cdots p_r^{a_r}, $$
with $$ p_1 < p_2 < \cdots < p_r. $$

First, ASSUME that one or more of the $a_i > 1.$ Now,
$$ \frac{ g(p_i)}{g(p_i^{a_i})} = p^{\delta - a_i \delta} = p^{\delta (1 - a_i)} < 1. $$
As a result, if we decrease that exponent to one, the value of $g$ is lowered, contradicting minimality. So
all exponents are actually 1.

Second, ASSUME that there is some gap, some prime $q < p_r $ such that $q \neq p_j$ for all $j,$ that is $q$ does not divide $n.$
Well, for real variable $x > 0,$ the function $$ \frac{x-1}{x^{1-\delta}} $$ is always increasing, as the first derivative is
$$ x^{\delta - 2} (\delta x +(1-\delta)). $$
It follows that, in the factorization of $n,$ if we replace $p_r$ by $q,$ the value of $g$ is lowered, contradicting minimality. So
the prime factors of $n$ are consecutive, beginning with 2, and $n$ is called a primorial.

Finally, what is the largest prime factor of $n?$ Beginning with 2, multiplying by any prime $p$ with
$$ \frac{p-1}{p^{1-\delta}} \leq 1 $$
shrinks the value of $g$ or keeps it the same, so in demanding the largest $n$ in case there are two attaining the minimum of $g,$ we take $n$ to be the product of all primes $p$ satisfying
$$ p - 1 \leq p^{1-\delta}, $$ or
$$ p^{1-\delta} \geq p-1 $$
as I first wrote it.

Examples are given in my first answer to this same question.

EEDDIITTTT: Jean-Louis Nicolas showed, in 1983, that the Riemann Hypothesis is true if and only if, for all primorials $P,$
$$ \frac{e^\gamma \phi(P) \log \log P}{P} < 1. $$
Alright, the exact reference is: Petites valeurs de la fonction d'Euler. Journal of Number Theory, volume 17 (1983), number 3, pages 375-388.

On the other hand, if RH is false, the inequality is true for infinitely many primorials and false for infinitely many. So, either way, it is true for infinitely many primorials (once again, these are $P = 2 \cdot 3 \cdot 5 \cdots p$ the product of consecutive primes beginning with 2).

For whatever reason, the criterion of Guy Robin, who was a student of Nicolas, got to be better known.