Determine if it is possible, for an entire function $f$, to vanish in the points $\{ \log k +i\log h : h,k \in \mathbb Z^+ \}$, and to have finite growth order.
Weierstrass' theroem about infinite products states there is an entire function $f$ vanishing in $\{ \log k +i\log h : h,k \in \mathbb Z^+ \}$, because, as $k,h$ increase, $|\log k +i\log h|$ goes to infinity. However I don't see how to prove that $f$ has finite growth order; I also know Hadamard's factorization theorem, but it doesn't seem to be useful because the finite order is in the hypothesis of the theorem, not in the thesis. Thanks in advance for any help