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Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function.
How can one show that there exists a constant $\theta>1$ such that $\dfrac{N_k}{\phi(N_k)} > \theta \log\log N_k$

My attempt: By Mertens' theorem, we know that $\theta \to e^{\gamma}$, hence it seems to follow that for sufficiently large $k$, we can take any $1<\theta<e^{\gamma}$ ?

In this question, the sign $''\to''$ was used to mean ''tends to'', as an example, for the Prime Number Theorem we write $\pi(x) \to \frac{x}{\log x}$, where $\pi(x)$ is the prime counting function, but obviously this is not the correct symbol, unfortunately, i'm yet to learn how to write it in MathJax. Any editing to the correct format will therefore be most welcome.

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  • $\begingroup$ I think you are spot on. But as you say this only establishes it for sufficiently large $k$. $\endgroup$ – Dan Brumleve Feb 12 '16 at 3:57

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