The maximum of the preimage of [1,x] through Euler's totient function A friend of mine and I have shown the following:
"For each $x \geq 1$ let $m := m(x)$ be the greatest positive integer such that $\varphi(m) \leq x$, where $\varphi$ is the Euler's totient function.
Then $m \sim e^\gamma x \log \log x$, as $x \to +\infty$, where $\gamma$ is the Euler-Mascheroni constant."
Both of us think that this should be a known result, but we found no references about it, so we ask here for them.
Thank you in advance for any references.
 A: So far, opinion is that I have not demonstrated much related to the given question. I will leave this here anyway, it's got references.
Let me begin with the explicit lower bound at LOWER,
$$ \phi(n) > \frac{n}{e^\gamma \log \log n + \frac{3}{\log \log n}}  $$ for $n>2.$  Apparently this is Rosser and Schoenfeld 1975.
ORIGINAL; It would seem you have the Nicolas criterion for RH, see item 83 at http://math.univ-lyon1.fr/~nicolas/publications.html  I have a reprint pdf that is better looking, if you want to email me. Just checked, the reprint on his site is two pages side by side photocopied from a book. Somehow I have it as it appeared in the Journal of Number Theory in 1983. 
There is an amusing postscript by Planat et al, http://arxiv.org/abs/1012.3613 
Having experimented by computer with various versions of these things, the Nicolas criterion is far easier to work with than Robin's or Lagarias'. Meanwhile, the procedure of Ramanujan for finding really extreme values of things, the two known sequences being his superior highly composite numbers and then the colossally abundant numbers, simply gives the primorials in this case. I wrote out a proof, if I can find it I will add a link. I think it was on MSE rather than here. 
I gave two answers here https://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below  and one is the proof of optimality for the primorials. 
