The following Euler product came up in some sieving applications:
$f(z, s) = \prod\_{\mbox{primes}} \left(1-\frac{z}{p^s}\right).$


What is known about this function? (Analytic continuation? Asymptotics?) This must be quite classical...



information on an Euler product
This function has indeed been studied before. For example, its reciprocal is
$$
f(z,s)^{-1} = \prod\_p \bigg( 1-\frac z{p^s} \bigg)^{-1} = \sum\_{n=1}^\infty z^{\Omega(n)} n^{-s},
$$
where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. When $|z| <2$, this function can be written as
$$
f(z,s)^{-1} = \zeta(s)^z g\_z(s)
$$
where the Dirichlet series $g\_z$ converges for $\Re s > \frac12$. In particular, for $|z|<2$ the function $f(z,s)$ can be analytically continued to $\Re s > \frac12$ minus a branch cut on $[\frac12,1]$, say. Similar things can be said for larger $|z|$ as well, although now the individual factors in the product defining $f(z,s)$ can vanish.


Information about this function appears in Section 7.4 of Montgomery/Vaughan's *Multiplicative Number Theory* and Chapter 5 of Tenenbaum's *Introduction to Analytic and Probabilistic Number Theory*; the key words "Selberg-Delange method" might also help you find references.


