Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

Question

I am really stuck on a homework problem, which boils down to the following:

We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions I know that satisfy such properties take their zeroes at integer values, but I have no idea how to adjust the picture to the set of values $n \text{ln}(n)$; apparently the inverse function for $x \mapsto x \text{ln}(x)$ is really complicated, so I am totally stuck.

I know that such a function exists, and I can give an infinite product representation for such a function, namely

$$\prod_{n=1}^{\infty} \Big(1 - \frac{z}{n\text{ln}(n)} \Big)\text{exp} \Big(\frac{z}{n\text{ln}(n)} \Big)$$

but the question asks for a closed form. Does anyone know how to start a proof?

Thanks a lot!

Answer 1

I think that $f(z)=0$ works pretty well but if you want a non constant one notice that $e^{n\ln(n)}=n^n \in \mathbb{N}$ and then you can use the fonction that you mentioned