Prescribe zeros of a meromorphic function

Question

Let $\{z_j\}_{j \geq 1}$ and $\{p_j\}_{j \geq 1}$ be sequences of distinct complex numbers. Set $$M(z)=1+\sum_{j=1}^{+\infty} \frac{r_j}{z-p_j}, \quad z \in \mathbb{C}.$$ Under which conditions we can find $\{r_j\}_{j \geq 1} \subset \mathbb{C}$ such that $M$ is meromorphic and $$M(z_k)=0, \quad \forall k \geq 1.$$ ? I guess it is linked to the Mittag-Leffler theorem but I don't see how to use it, nor if it gives sharp conditions.

Answer 1

The Mittag-Leffler theorem says that there exists a meromorphic function with principal part $r_j/(z-p_j)$ at $p_j$; it does not say that the funtion is given by that sum. (And in fact the conclusion of ML still holds when that sum does not converge.)

It's clear that for any sequence $p_j$ there exists a sequence $r_j>0$ such that the sum converges to a meromorphic function.