TITLE: Euler product over a finite subset of composite numbers
QUESTION [6 upvotes]: The product $$\prod_{p\text{ prime}}\frac{p^n}{p^n-1}=\zeta(n)$$ is well-known. This works over primes, and only over the entire set of them.
Is any similar expression known for composites? For a finite subset of them?
e.g. $$\prod_{c\text{ composite, c}=c_{i}}^{c=c_{j}}\frac{c^n}{c^n-1}=f(m)$$ for some $f$.
This is generic. If such an expression is not known, is anything known for the case $i=1$ and $n=1$? If not, then for $i=n=1$ and $j=\infty$? If that is also not known, is anything known on $\lim \inf f(m)$ or $\lim \sup f(m)$?

REPLY [2 votes]: This Answer addresses the coefficients of the Dirichlet-Series of $f(s) = \prod_{c \text{ comp}} \frac 1{1-c^{-s}}$.
We introduce
$$g(s) = \prod_{n=2}^\infty (1 + n^{-s} + n^{-2s} + \dots ),$$ which is a holomorpic function on $\Re s > 1$, as $\left | 1 - \frac 1{1-n^{-s}}\right | \ll n^{-s}$.
This can be re-written as a dirichlet-series
$$g(s) = \sum_{n=1}^\infty c(n) n^{-s},$$
where $c(n) = \#\{(e_2, e_3, \dots) \in \mathbb N_{\geq 0}^* | 2^{e_2} 3^{e_3} 4^{e_4} \dots = n \}$ is the amount of multiplicative partitions of $n$. With $\mu$ (the möbius-function) we obtain
$$\prod_{c \text{ comp}} \frac 1{1-c^{-s}} = \zeta(s)^{-1} g(s) = \sum_{n=1}^\infty a(n) n^{-s},$$
where $a(n) = (c \star \mu)(n) = \sum_{d \mid n} \mu(d) c(\frac nd)$.