I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski.
They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\Lambda_f(x) = \kappa\log x+O(1)$$ where $\Lambda_f(n)$ is given by corresponding Dirichlet series $-D'_f(s)/D_f(s)$, $\kappa>-1/2$ is a constant, with $$\sum_{n\le x}|f(n)|\ll(\log x)^{|\kappa|},$$ then the Euler product $$\frac{D_f(s)}{\zeta(s+1)^{\kappa}} = \prod_{p}\left(1-\frac{1}{p^{s+1}}\right)^\kappa\left(1+\frac{f(p)}{p^s}+\frac{f(p^2)}{p^{2s}}+\cdots\right)$$ has limit $$\prod_{p}\left(1-\frac{1}{p}\right)^\kappa\left(1+f(p)+f(p^2)+\cdots\right)$$ as $s\to 0+$, without further explanation (it is previously known that the function $D_f(s)/\zeta(s+1)^\kappa$ tends to a limit as $s\to 0+$).
But I cannot see why. Below is my trial.
(1) By means of Abel summation, it is known that if a Dirichlet series $D_g(s)$ converges at $s=s_0$, then it is uniformly convergent in the sector $\{s\in\mathbb{C}|\textrm{Arg}(s-s_0)<\theta\lor s=s_0\}$ for a fixed $\theta\in(0,\pi/2)$. Hence if we can show that the given Euler product converges absolutely(in the sense that we take absolute value for inner series, too), then we can write it as a corresponding Dirichlet series, and hence use continuity of the Euler product to conclude. But I couldn't show that the product converges at $s=0$. I don't even know the series $1+f(p)+f(p^2)+\cdots$ converges or not.
(2) I took logarithmic derivative of the quotient and see if the corresponding Dirichlet series converges (although it may not have direct relation...). Then we have $$-\frac{D'_f}{D_f}(s)+\kappa\frac{\zeta'}{\zeta}(s+1) = \sum_{n=1}^{\infty}\frac{\Lambda_f(n)-\Lambda(n)/n}{n^s},$$ where the coefficient has partial sum of growth $O(1)$ by the first condition (and by Mertens theorem $\sum_{n\le x}\Lambda(n)/n = \log x+O(1)$). This does not help too, since it only says that the series converges for $\Re s>0$. Or by Abel summation this says that the LHS above is bounded near $s=0$, but this is not enough as we don't have any information about possible continuation of $D_f(s)$ beyond the half-plane $\Re s>0$.
I am aware of the duplicate Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book , but this does not answer my question, and it's too old post, so I cannot expect further answer even if I have written a comment. If you want, I can list my doubts on the answer above.
