Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2008 printed variant / 6 / ii

Here the zeros accumulate only at the boundary point zero. An explicit holomorphic, nonzero example on the upper half-plane isAt $z_n=i/(n+1)$ its argument is $\pi (n+1)$, so it vanishes. It is holomorphic because the only excluded point of its formula is zero; it is not identically zero, for instance at $z=2i$. Thus the answer is yes.