zeros of entire functions Consider the product
$$\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$
I've proven that this product converges uniformly on compact subsets of complex plane since the serie $\sum_{n=0}^{+\infty}|\frac{e^{2\pi iz}}{e^{2\pi n}}|$ does.
Now i'm interested to zeros of $F$, the entire function to which the product converges. How can i find them? Can i say that all the zeros of $F$ are those complex numbers $z$ such that $e^{-2\pi n}e^{2\pi i z}=1$? If yes, zeros are of the form
$$z_{n,k}=-in+k$$
$n,k$ integers. Do you think it's correct?

Yes, by definition of convergence for products, we still have that the product is $=0$ iff one of the factors is $=0$
