I have troubles in understanding the proof of the criterion which states that an infinite product exists iff the series of the complex logarithms of the terms of the product converges.
In particular, the proof says that if the infinite product converges then for every $\varepsilon >0$ exists $N$ natural s.t. for $p\geq m>N$ we have $\vert \prod_{n=m}^p a_n -1 \vert< \varepsilon$; so for $\varepsilon$ small enough $\vert\arg(\prod_{n=m}^p a_n)\vert<\pi/2$, so $\log(\prod_{n=m}^p a_n)=\sum_{n=m}^p \log(a_n)$ tends to zero for $m,p \to +\infty$, hence the series of the principal logarithms satisfies the Cauchy convergence criterion.
My doubt is: why for the complex logarithm case the logarithm of the product is the sum of the logarithms? Shouldn't be there a correction term on the sum of the arguments? In which cases the complex principal logarithm of a product is the sum of the principal logarithms?
Thanks a lot!