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!

The function
$$
z\mapsto\frac{z-1}{z+1}
$$
maps the right half plane to the unit circle. The points $\sqrt{n}$ get mapped to
$$
\frac{\sqrt{n}-1}{\sqrt{n}+1}
$$
Since
$$
\sum_{n=1}^\infty\left(1-\left|\frac{\sqrt{n}-1}{\sqrt{n}+1}\right|\right)=\sum_{n=1}^\infty\frac2{\sqrt{n}+1}=\infty
$$
the Szegö Theorem says that the only bounded function on the unit disk with those roots vanishes identically.
