Divisors of meromorphic functions on compact Riemann surfaces

Question

If $X$ is a compact Riemann surface and $f$ a meromorphic function on $X$ let $(f)=\sum_i n_iz_i-\sum_jm_jp_j$ be its divisor. If $g$ is another meromorphic function on $X$ and $(f)=(g)$ is it true that $f=cg$ where $c$ is a constant? Or is it true only if $X$ is the Riemann sphere?

Answer 1

It's true for all compact Riemann surfaces.
Indeed, $div (f/g)=0$ so that $f/g$ is holomorphic and without zeros, hence is a non-zero constant.