in the book The Prime Numbers and Their Distribution from Tenenbaum is a note about the existence of a complex continuation of the logarithm:
Let $\alpha>0$ and an analytic function $f(s)$ with no zeros for $Re(s)>\alpha$ and $f(s)$ is real and positive for $s \in (\alpha,\infty)$. then $\log(f(s))$ is defined for $Re(s)>\alpha$ and $Re(\log(f(s))=\log(|f(s)|)$. But there is no proof in book.
I would like to proof it, I know:
Let $f:D \longrightarrow \mathbb{C}$ is analytic and $D$ is simple connected domain, $f(s) \neq 0$ for all $s \in D$. Then there is an analytic funtion $h: D \longrightarrow \mathbb{C}$ with the property
$f(z)=\exp(h(z))$
so $h$ is an analytic branch of logarithm of $f$. But how I can obtain the property $Re( \log(f(s))= \log(|f(s)|)$?
The motivation is to get a logarithm of the riemann $\zeta$ function which is real for all $s>1$.