Can a holomorphic function vanish on an infinite 2D rectangular grid? Consider a set of values $x_{n,k}=n+ik$, where $n,k\in\mathbb Z$. Does there exist a non-constant holomorphic function $f$, such that $f(x_{n,k})=0$ for all $n,k$?
I've tried to construct such a function as e.g. a product of $(x-n-ik)$, but it rapidly diverges as I increase maximum values of $|n|$ and $|k|$, and I can't even suppress its growth by a gaussian or anything like that — it's too non-uniform, so in the limit it'd become identical zero. I've also considered something like $\sin(\pi x)\sinh(\pi x)$, but this only vanishes on a "cross" of points instead of the whole grid.
So I suppose the answer is no, but how to (dis)prove it? And if there do exist such functions, what would be a concrete example?
 A: For every lattice there is a non-zero entire function that vanishes precisely on this lattice. Very useful examples of this are theta functions. Here is one way to express such a theta function. Let $q$ be a complex number and $0 < \lvert q \rvert < 1$. For $z\in\mathbb{C}^{\ast}$ the product $$\theta(q; z)=\prod_{k=0}^{\infty}(1-q^kz)(1-q^{k+1}z^{-1})$$ defines a holomorphic function in $z$ which has a simple zero precisely for $z=q^m$ with $m\in\mathbb{Z}$. Then $\theta(q; e^{2\pi\mathrm{i}z})$ is entire with simple zeroes on a lattice $\mathbb{Z}+\mathbb{Z}\tau$ for some $\tau$ in the upper half plane such that $q=e^{2\pi\textrm{i}\tau}$.
A: The Wierstrass Elliptic functions are "doubly periodic" meaning that they are defined on some parallelogram in the complex plane and then repeat on a lattice of those parallelograms.  There are elliptic functions for which the "period" is a square oriented on the real and imaginary axes.
The elliptic functions are entire and you can find one with arbitrary values at the corners of the parallelogram. So the answer to your question is yes, an entire function can vanish on an infinite 2D grid.
