Question

Construct an entire function f in complex with simple zeros at an = n^(1/2) for n belongs to natural numbers and no other zeros

Answer 1

By the Weierstrass factorizaion theorem we obtain that $f(z)=\prod\limits_{n=1}^{\infty}E_n\left(\frac{z}{a_n}\right)$ is a entire function, such that $\{z: f(z)=0\}=\{a_n\}_{n\in\mathbb N}$, and these zeros are simple, where $E_n(z)=(1-z)\exp\left(\sum\limits_{k=1}^n \frac{z^k}{k}\right)$

Indeed

1) $\lim\limits_{n\to\infty}|a_n|=+\infty$

2)For every $r>0$ there is $N$ such that $\left|\frac{r}{a_n}\right|<\frac{1}{2}$ for all $n>N$, then$\sum\limits_{n=1}^{\infty}\left|\frac{r}{a_n}\right|^{n+1}=\sum\limits_{n=1}^N\left|\frac{r}{a_n}\right|^{n+1}+\sum\limits_{n=N+1}^{\infty}\left|\frac{r}{a_n}\right|^{n+1}\le$

$\le\sum\limits_{n=1}^N\left|\frac{r}{a_n}\right|^{n+1}+\sum\limits_{n=N+1}^{\infty}\left(\frac{1}{2}\right)^{n+1}$, so $\sum\limits_{n=1}^{\infty}\left|\frac{r}{a_n}\right|^{p_n+1}$ , where $p_n=n$, is convergent series.

Then $f(z)=\prod\limits_{n=1}^{\infty}E_{p_n}\left(\frac{z}{a_n}\right)=\prod\limits_{n=1}^{\infty}E_n\left(\frac{z}{a_n}\right)$ is a entire function with simple zeros $\{a_n\}_{n\in\mathbb N}$.

Answer: $f(z)=\prod\limits_{n=1}^{\infty}E_n\left(\frac{z}{a_n}\right)=\prod\limits_{n=1}^{\infty}E_n\left(\frac{z}{\sqrt{n}}\right)$, where $E_n(z)=(1-z)\exp\left(\sum\limits_{k=1}^n \frac{z^k}{k}\right)$