Has anyone characterized the zeroes of the Bell numbers?

Question

I was reading this post about the Bell Numbers where users Lucian and Vladimir Reshetnikov give us Dobiński's formula for the Bell numbers

$$ B(x) = \frac{1}{e} \sum_{k=1}^{\infty} \frac{k^x}{k!}. $$

Now I was trying to reason about this function on the complex plane. It's easy to reason that for no value of $x$ such that $|x| < \infty$ will this expression have a singularity so this function definitely doesn't have any poles.

Now the other question I wanted to turn my attention to are, can we characterize the zeroes of this function? Are there only finitely many or countably many? Is there some natural characterization of the zeroes (like they all live on some well known curve(s))?

There also appears to be some Riemann surface structure attached to this wherein $k^x$ is multivalued for complex $x$ as it is $e^{\ln(k)x}$ where the choice of branch of $\ln(k)$ needs to be made.

Answer 1

The function $B(z)$ is an example of an almost periodic function. The zeroes of an almost periodic function that is holomorphic on some strip are also almost periodic, so such a function either has no zeroes at all or infinitely many zeros.

Let $f(z)=\sum_{k=0}^{\infty}a_ke^{b_kz}$ where each $b_k$ is real and each $a_k$ is complex. Suppose that $\alpha,\beta\in[-\infty,\infty]$ and that the sum converges uniformly and absolutely on compact subsets of the strip $\{z:\alpha<\text{Re}(z)<\beta\}$.

Now, there is a sequence of real numbers $r_n$ with $r_n\rightarrow\infty$ where $$\lim_{n\rightarrow\infty}(e^{ir_nb_k})_{k=0}^{\infty}=(1)_{k=0}^{\infty}.$$

This ensures that $f(z+ir_n)\rightarrow f(z)$ and $f(z-ir_n)\rightarrow f(z)$ uniformly on compact subsets of $\{z:\alpha<\text{Re}(z)<\beta\}$. Therefore, by Hurwitz' theorem, the locations of the zeroes of $f(z+ir_n)$ will converge to the locations of the zeros of $f(z)$.