This is the question from Lectures on Riemann surfaces (Otto forster), exercise 10.3:
Suppose $X$ is a Riemann surface and $\omega$ is a meromorphic 1-form on $X$ which has residue zero at every pole. Show that there is a covering $p: Y \to X$ and a meromorphic function $F$ on $X$ such that $dF=p^*w$.
Thanks.