# Equivalence of the term "divisor"

In the theory of Riemann surfaces, for a meromorphic function $$f$$ on a Riemann surface $$X$$, we define the divisor of $$f$$ to be the function $$D$$ which maps $$f \longmapsto \sum_{p \in X} \text{ord}_p \cdot p,$$ where $$\text{ord}_p$$ denotes the order of the root or pole of $$f$$ at $$p$$.

In the context of Stein manifolds however, we define a divisor to be an element $$d \in H^0(X, \mathscr{D})$$, where $$\mathscr{D}$$ is the quotient sheaf of $$\mathscr{M}^{\times}$$ by $$\mathscr{O}_X^{\times}$$, where $$\mathscr{M}^{\times}$$ is the sheaf of germs of invertible meromorphic functions and $$\mathscr{O}_X^{\times}$$ is the sheaf of germs of invertible holomorphic functions.

Are these definitions equivalent? If they are, I can't see why.

Note that this does seem like a question that should be posted on Math.StackExchange, but I have found little to no luck asking questions of anything north of the elementary theory of analytic functions of several complex variables on that site.