Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book

Solution 1:

Notice that $\prod_{p\leq\\z}(1-p^{-1})^{k}\left(\sum_{\nu=0}^{\\r}f(p^{\nu})\right) \leq O((\log z)^{-k})\left(\sum_{n\leq\\x}f(n)\right)$, where $\\n\leq\ x$ is chosen accordingly to $\\z\geq\ p$. So, letting $\\x,z\to \infty$ we note that the product is convergent. Now, $\prod_{p}(1-p^{-s-1})^{k}\left(\sum_{\nu=0}^{\infty}f(p^{\nu})p^{-\nu s}\right) = D_f(s)/(\zeta(s+1))^k$ which is absolutely 'okay' for $\\Re(s)>0$, also $D_f(s)/(\zeta(s+1))^k$ is analytic at $s=0$ by checking limit (having no pole there) . Hence, by well known facts from complex analysis letting $\\s\to\ 0+$ we get the result.