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Up to which value of $k$ has this been proved true?

$$\frac{N_k}{\phi(N_k)} > e^\gamma \log\log N_k$$

Thank you.

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    no idea. This is the criterion of Nicolas, and is the easiest with which to experiment. It is not necessary to produce the primorial $N_k,$ just the product of the $p / (p-1)$ the way you wrote it, along with the sum of $\log p$ http://math.univ-lyon1.fr/~nicolas/petitsphi83.pdf I did it for the first few dozen primes, I guess, easy enough. – 2017-02-12
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    see also https://arxiv.org/abs/1012.3613 although, once again, no mention of anyone doing computations; mine fit on one page – 2017-02-12

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from comments, the two references I know are

http://math.univ-lyon1.fr/~nicolas/petitsphi83.pdf

https://arxiv.org/abs/1012.3613

I counted lines, proved up to $k=47$

Turns out I saved my computations as a jpeg. Notice that I'm not proving much, just calculating the number for primes up to 211. This is what you asked about. I do not know of anyone who has extended this computation, which would require better decimal accuracy

enter image description here

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    Thank you very much! I will wait to see if anyone has a result for larger $k$ – 2017-02-12
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    @user3141592 fairly easy program, you should try it yourself – 2017-02-12