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Let $\varphi$ be the Euler totient function, $N_k$ denote the product of the first $k$ primes and define $$f(k, r) = \frac{(N_k)^r}{\varphi((N_k)^r) \log \log ((N_k)^r)}.$$

where $r \in \mathbb{N}$. It is a classical result of Mertens that $\limsup_{k \rightarrow \infty} f(k, 1) = e^{\gamma}$ where $\gamma$ denotes the Euler-Mascheroni constant. Is this also true for every positive integer $r$ ?

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  • $\begingroup$ I took the liberty to edit your question to remove a supernumerary opening parenthesis. $\endgroup$ Jan 18, 2021 at 13:23
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    $\begingroup$ Rewriting the definition, one gets $f(k,r) = f(k,1)\cdot \frac{\log\log N_k}{(\log\log N_k)+(\log r)}$ so the result follows directly I think. $\endgroup$ Jan 18, 2021 at 14:42

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