Mertens' third theorem states that:
$$\lim_{n \to \infty} \log{n} \prod_{p\le n} \frac{p-1}{p} = e^{-\gamma}$$
For $p$ a prime number. Is it possible to generalise that result when the product is taken over the integers $q$ with exactly $k$ different prime factors ($\omega(q)=k$)?
$$\prod_{q\le n} \frac{q-1}{q} \sim \ ?$$
With $\omega(q)=k$. Mertens' theorem would be the case $k=1$.
I have not found any good reference about this subject, so any information is welcomed. Thank you in advance
See how it works for the case $k=2$, $$\sum_{n\ge 1,\omega(n) = k}n^{-s-1} =\frac12 (\sum_{p^m} (p^m)^{-s-1})^2 - \frac12 \sum_{m,l,p} (p^{l+m})^{-s-1}=\frac12 \sum_{p^m} (p^m)^{-s-1}+F_k(s)$$ where $F_k(s)$ converges for $\Re(s) > -1/2$,
which gives the asymptotic $$\sum_{n\le x, \omega(n)=k}n^{-1} = A(\log\log x)^2+B\log\log x+C+O(\frac1{\log\log x})$$ $$\prod_{n\le x,\omega(n)=k} (1-n^{-1}) =\exp(- A(\log\log x)^2-B\log\log x-C+D+O(\frac1{\log\log x}))$$ then use induction on $k$ repeating the same idea.