Let $X$ be a Riemannian surfaces with a divisor $D$ and let $E$ be a holomorphic complex vector bundle of rank r on $X$.
1) The Riemann-Roch theorem is used to give an estimate of the dimension of the vector space of the holomorphic sections of $E$, i.e
where g(X) is the genus of $X$.
Here my question: let $D$ be as above, is it possible to write a version of the above formula that gives informations about the dimension of the vector space of meromorphic sections of $E$ with pole in $D$? (this is possible for line bundle thanks to the the correspondence line bundles---divisors, for this reason I'm expecetd something involve the determinant bundle of $E$)