On a black board in the floor of my university was this question as a puzzle, and I would like to know why the $\exp(g(z))$ in the Weierstrass Product is constant $1$, in the question one answer refers to a comment (of a user which probably changed his name ) in which is a link but the link seems broken.

Someone wrote that because of the function is in $\mathcal{O}(\exp(|z|)$ it follows that $g(z)$ is constant, but the function is only in $\mathcal{O}(|z|\cdot \exp(|z|)$ and I don't see how the asymptotic should help, I guess he want to make something like Liouville, but this is only a guess.