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 $r$ ?