Do you know any nontrivial analytic function f(z) with zeros only at positive integer values of the argument z = 1, 2, 3, 4, ... ? If yes, please give some example.
PS: I already thought of $f(x)=\frac{1}{\Gamma(-x+1)}$. Any other nice options?
EDIT: To avoid trivial solutions due to restriction of definition range, please consider the required function to be defined in the whole complex plane.