Suppose $f$ is an entire function with the property that
$|f(z)|\leq C\exp(|z|^\rho)$ with $\rho\geq0$. And $f(\log(n))=0 \forall n\geq3 $
Now I have to prove that $f$ is identically $0$.
I was thinking that if $f$ could be shown to be uniformly bounded then by Liouville's theorem we'd have that it is identically $0$.But I couldn't see how to use $f(\log(n))=0$.
Can someone please help me solve this ? Note that this is a problem from the chapter Riemann mapping theorem.