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?
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 $q=e^{2\pi\textrm{i}\tau}$.