`  :I don't know, I'd have to go to the source. I may have read a review I had found on arxiv, not sure. I can certainly use intuition to guess: emptiness) I don't think it matters: if the statement holds for interior of U non-empty, then one has, as a lemma, the special case of a collection of points (i.e. where int U is empty), right?   :Zeros) Intuition says that a finite number of zeros shouldn't matter: for a holomorphic function that has a finite number of zeros, I can find another holomorphic function that approximates it arbitrarily well on U, but has no zeros in U (right? I think so ..) A similar statement holds true, for a countable number of zeros, although that is clearly ``harder``. Right? Finally, there are no holomorphic functions that have an uncountable number of zeros, right? So, by this lemma, the question of zeros doesn't much matter.  : But I find what I wrote strange, since, if that is the case, then is Voronov saying that the Riemann zeta does everything except get arbitrarily ``flat``, on any U? So I'm not sure what the ``non-vanishing`` means.   :Of course, its late at night, I might be spouting nonsense. Caveat Emptor.   `