Consider the product
$$\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$
I know that this product defines an entire function $F$. I must show that the order of growth of $F$ is finite, at most 2.
My definition of order of an entire function is
$$\rho=\inf\{\lambda>0: \displaystyle\sup_{|z|=r} |f(z)|=O(e^{r^{\lambda}}),r\rightarrow\infty\}$$
Can someone give me an help?
Best Answer
The zeros of $F$ are $n+ki$ for $n=1,2,\dots$, $k\in\mathbb{Z}$. If $p>2$ then $$ \sum_{n=1}^\infty\sum_{k=-\infty}^{\infty}\frac{1}{|n+ki|^p}<\infty. $$ This shows that the order of $F$ is at most $2$.