Question

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...

Answer 1

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.