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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.

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    $\begingroup$ It seems this is a 1903 theorem of Landau. See math.stackexchange.com/questions/323144/… $\endgroup$ – S. Carnahan Sep 9 '14 at 8:48
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    $\begingroup$ Landau’s result only gives $$\limsup_{x\to+\infty}\frac{m(x)}{x\log\log x}=e^\gamma,$$ or am I missing something? $\endgroup$ – Emil Jeřábek 3.0 Sep 9 '14 at 9:09
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    $\begingroup$ @S.Carnahan Emil Jeřábek is right. Landau's theorem only gives $m(x) \leq (e^\gamma + o(1)) x \log \log x$, as $x \to +\infty$. $\endgroup$ – user40023 Sep 9 '14 at 9:46
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    $\begingroup$ Choose the largest primorial below $\sqrt{x}$, and find a multiple of this close to $e^{\gamma} x\log \log x$. That does the job (together with the classical lower bound for $\phi(n)$). $\endgroup$ – Lucia Sep 9 '14 at 19:37
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    $\begingroup$ @Lucia: I think your comment would make a fine answer. $\endgroup$ – Emil Jeřábek 3.0 Dec 8 '14 at 22:13
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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.

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  • $\begingroup$ What is the connection between Nicolas criterion and the question? $\endgroup$ – Emil Jeřábek 3.0 Sep 9 '14 at 17:41
  • $\begingroup$ @Emil, his wording is backwards of Nicolas; screw it, given $0 < \delta < 1,$ the global minimum of $\phi(n) / n^{1-\delta}$ occurs at a specific primorial. Let me think... alright, so the largest number to have $\phi $ as small as $C n^{1-\delta}$ is a primorial, which one depends on $\delta$ $\endgroup$ – Will Jagy Sep 9 '14 at 17:45
  • $\begingroup$ No, I still don’t understand your argument. The property claimed in the question is that for every $\delta>0$ and all sufficiently (depending on $\delta$) large $n$, there is $m\ge(e^\gamma-\delta)n\log\log n$ such that $\phi(m)\le n$. Constant powers of $n$ do not appear anywhere. Since the inequality implies $m\le(e^\gamma+o(1))n\log\log n$, these $m$ cannot be primorials, they are much more densely distributed. $\endgroup$ – Emil Jeřábek 3.0 Sep 9 '14 at 18:12
  • $\begingroup$ @emil, alright; I will leave it there anyway, as it remains information that the OP seems not to know. In any case, he can read Rosser and Schoenfeld 1975 and Nicolas 1983 and know a good deal more than I do. $\endgroup$ – Will Jagy Sep 9 '14 at 18:15

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