Clunie–Eremenko–Rossi conjecture on zeros of sums of Cauchy kernels

Let $c_k>0$, $t_k\in\mathbb{C}$, and

$$\sum\limits_{k\in\mathbb{N}}\frac{c_k}{|t_k|}<+\infty.$$

Define the meromorphic function

$$f(z)=\sum\limits_{k\in\mathbb{N}}\frac{c_k}{z-t_k}.$$

Clunie–Eremenko–Rossi conjecture. The function $f(z)$ has infinitely many zeros.

The claim concerns the zero sets of meromorphic functions formed from sums of Cauchy kernels with positive coefficients and summable weighted pole locations. Although the parser's status is marked as disproved, the supplied status evidence states that as of 2023 the conjecture remains neither proven nor disproven; this discrepancy should be checked against the source and current literature.