# On the definition of divisors in Riemann Surfaces

The sum notation for a Divisor $D$ in a Riemann Surface $X$ (as in Miranda's "Algebraic Curves and Riemann Surfaces") is $$D=\sum_{p\in X} D(p)\cdot p$$ That is, $D$ assumes the value $D(p)$ at $p$. For example, a principal divisor of $f$ is the divisor $$div(f)=D=\sum_{p\in X} ord_p(f)\cdot p$$

which, if I understood it correctly, means that this divisior is the function $$\begin{array}{rcc} div(f):&X&\rightarrow&\mathbb{Z}\\ &p&\mapsto &ord_p(f) \end{array}$$

Now, defining the divisor of zeroes and the divisor of poles as $$div_0(f)=\sum_{p\text{ with }ord_p(f)>0} ord_p(f)\cdot p$$ and $$div_{\infty}(f)=\sum_{p\text{ with }ord_p(f)<0} (-ord_p(f))\cdot p$$

respectively, how are those functions defined on the points $x\in X$ such that $ord_x(f)\leq 0$ (in the first case) or $ord_x(f)\geq 0$ (on the second)? The question arises since the divisor must be defined in the whole Riemann Surface $X$.

For points $p$ that don't appear in the summation defining the divisor of zeros (resp. the divisor of poles) the value of the divisor is zero. This ensures that $\mathrm{div}(f)=\mathrm{div}_0(f)-\mathrm{div}_\infty(f)$.
In general, when working with a free abelian group on some set $X$ (the Riemann surface in your example), if one writes down an expression which sums over a subset of $X$, it is to be understood that the coefficients corresponding to the other elements of $X$ are all zero.