Bounded Holomorphic function on Right half plane. Does there exist a Bounded Holomorphic function defined on Right half plane which have all $\sqrt{n}$ as root for all natural number $n$?
I guess It is a just $0$ function.
But How Could I approach this one?
(I've been trying to use Blascke product.)
Thanks!

If $f$ is a bounded non-constant holomorphic function on the disc then $\{1-|z| \mid f(z)=0\}$ is summable (see here). The function $$z\mapsto\frac{1+z}{1-z}$$ maps the unit disc onto the right half plane. The pre-image of $\sqrt{n}$ is $$\frac{\sqrt{n}-1}{\sqrt{n}+1} = 1 -\frac2{\sqrt{n}+1}$$
so a bounded function on the unit disc with those roots must be identically zero.  Therefore also a bounded function on the right half plane with roots $\sqrt{n}$ must be identically zero.
