About the determination of complex logarithm Although it must be a silly question, I am really confused.
For complex logarithm, in general, $$\log(z_1z_2)\neq\log(z_1)+\log(z_2)$$ even if the logarithm is already determined. I think this is true, right?
However, in some proof from complex analysis, one uses logarithm to let a product (even infinite product) be a summation without considering any determination of logarithm. Is that correct?
For example:
We know that $$\prod_{p}\left(1-\frac{1}{p^s}\right)^{-1}=\sum_{n=1}^\infty\frac{1}{n^s}=\zeta(s)$$
holds for $Re s>1$. In some books, the author states that by 
$$\log\zeta(s)=-\sum_p\log\left(1-\frac{1}{p^s}\right)$$
Is that correct?
Sometimes, I think complex logarithm is really "annoying", because I always have to worry about the determination of it. Do I have to worry about that?
Thank you very much!

Much like the inverse trig functions, the logarithm may take on an infinite set of values:
$$\log{z} = \log{|z|} + i (\theta + 2 \pi k) $$
where $k \in \mathbb{Z}$ and $\theta = \arg{z}$.  In this sense, you are right.  Nevertheless, if we define a principal branch of $\log{z}$ such that
$$\arg{z} = \mathrm{Tan}^{-1}{\frac{\Im{z}}{\Re{z}}}$$
where $\mathrm{Tan}^{-1}$ refers to the principal branch of the $\tan^{-1}$ function, with a range of $[-\pi,\pi)$, we may then define a unique value of the logarithm of a complex number.  Some use the notation $\mathrm{Log{(z)}}$ to denote this principal value such that $\Im{\mathrm{Log{(z)}}}$ takes on a unique value.  In this sense, then, $\mathrm{Log{(z_1 z_2)}} = \mathrm{Log{(z_1)}} + \mathrm{Log{(z_2)}}$.
