TITLE: Have you ever seen this product?
QUESTION [12 upvotes]: Given $k$, what is the value of the following product?
$$\prod_{p\textrm{ prime}}\left(1-\frac{k}{p^2}\right).$$
Clearly for $k=1$, we have $\zeta(2)^{-1}$ (where $\zeta(s)$ is the Riemann Zeta function).

REPLY [5 votes]: Elementary comment. For a fixed real number $x$, define a multiplicative function $f(x;n)$ as follows. Factor $n=p_1^{e_1}\cdots p_r^{e_r}$ and set $f(x;n)=x^{e_1+\cdots+e_r}$. Then
$$
  \prod \left(1-\frac{x}{p^s}\right)^{-1} = \sum_{n=1}^\infty \frac{f(x;n)}{n^s}.
$$
The function $n\to e_1+\cdots+e_r$ that counts the number of prime divisors of $n$ with multiplicity must come up in lots of places, but I don't recall seeing it appear like this in a Dirichlet series.