Elliptic divisor-sum identity (${\sum }_j{r}_j{z}_j-{\sum }_l{m}_l{p}_l\in \Lambda$)

For a nonzero elliptic function, the weighted sum of its zero positions minus its pole positions belongs to the period lattice. Integrating $z{f}^{\prime }/f$ around a fundamental parallelogram proves this: translating opposite edges gives a lattice-linear combination of integrals of $f^{\prime }/f$, each an integer multiple of $2\pi i$. The unweighted integral proves that the zero and pole multiplicities have equal total.