Taking complex logarithm of some multiplicative identities

Question

Let $\log(\cdot)$ be the principal value complex logarithm. We know that, given $z_1, z_2\in \mathbb{C}\setminus\mathbb{R}_{\le 0}$, the identity $\log(z_1z_2)=\log(z_1)+\log(z_2)$ can fail. Nevertheless, in some complex analysis textbooks I've read the following implications: $$\zeta(s)=\prod_p \frac{1}{1-p^{-s}}\implies \log \zeta(s)=\sum_p \log\left(\frac{1}{1-p^{-s}}\right)$$ $$\Gamma(z)=\lim_{n\to\infty}\frac{n^z n!}{z(z+1)\cdots(z+n)}\implies \log\Gamma(z)=\lim_{n\to\infty}\left[z\log n-\log(z+n)-\sum_{k=1}^n\log\left(1+\frac{z-1}{k}\right)\right].$$ These are only two examples. In general, how is one supposed to interpret such identities? Are they equalities between multi-valued functions?

Answer 1

I think the usual way one interprets these identities is as integrals of logarithmic derivatives: i.e., $\Gamma'(z)/\Gamma(z)$ has a Mittag-Leffler expansion, and $\log{\Gamma(1)}$ can be chosen to be $0$ unambiguously, and then $\log{\Gamma(z)} = \int_1^z \frac{\Gamma'(w)}{\Gamma(w)} \, dw$ is defined by integrating term-by-term, which is well-defined on a simply-connected domain including $1$ (in $\Gamma$'s case, $\mathbb{C} \setminus \mathbb{R}_{\leq 0}$, and obviously a rather more complicated one for $\zeta$).