Quote from Grauert & Remmert's Theory of Stein spaces: 'Behnke and Stein showed in 1948 that the Mittag-Leffier Partial Fraction Theorem and the Weierstrass Product Theorem (i.e. the Cousin Theorems) are valid on non-compact Riemann surfaces. The following lemma appears at the end of their paper:

Hilfssatz C: Let $D$ be a discrete set in a non-compact Riemann surface $X$. For every $p\in D$ let $z_p$ be a local coordinate at p. Suppose that at all $p \in D$ there is prescribed a finite Laurent-Taylor series $h_p = \sum_{\nu=-m_p}^{n_p}a_\nu z_p^{\nu}$, $0\leq m_p,n_p<\infty$. Then there exists a function $H$ which is meromorphic on $X$, holomorphic on $X\setminus D$, and whose Laurent development at $p$ with respect to $z_p$ agrees with $h_p$ up to the $n_p$-th term.'

Does $H$ still exist if we ask it to have no zeroes in $X\setminus D$? That is, I want to $H$ to realize *exactly* a prescribed divisor. References are welcomed.

Coherent Analytic Sheavesby GR, or Forster's book on Riemann surfaces. $\endgroup$