Can $\zeta(s)$ be written in the form $\zeta(s)=f(s)+g(s) i $ for some subset of $\mathbb{C}$? I mean, is it possible to develop at least one of the formulas of $\zeta(s)$ so you get something like...
$$ \zeta(s) = \Re(\zeta(s))+i·\Im(\zeta(s)) \; \; \\ \text{or} \\ \zeta(s) = \Re(\zeta(s))+i·\Im(\zeta(s))+ (\text{some-complex-function-that-you-can-keep-solving}) $$
Being $\Re(\zeta(s)):\mathbb{C} \rightarrow \mathbb{R}$ and $\Im(\zeta(s)):\mathbb{C} \rightarrow \mathbb{R}$ both real functions. Concerning the domain of the function, I am not really interested in all complex numbers: just if there is a subset that allows this rearrangement of the formula. I just let you a clear analogy:
$$ e^{ix}=\text{cos}x+i·\text{sin}x $$
This is the thing that I want to know. I've been looking in internet (this is not my field) but I haven't found anything about this- I'm still looking for an explicit representation of the real and imaginary part of the $\zeta(s)$ function as real real functions. Could anybody help me?? Thanks!!