The Weierstrass’ problem in one complex variable asked, if there is holomorphic function with the pre-assigned zeros in $\Bbb{C}$? This is proved using the Weierstrass infinite product.
The second Cousin problem ask given open covering $\{U_i\}$ of some region $\Omega \subset \Bbb{C}^n$, and $f_i \in \mathcal{O}(U_i)$ such that $f_i/f_j$ has no zero on $U_i\cap U_j$ and holomorphic, exist $f\in \mathcal{O}(\Omega)$ such that $f_i/f$ holomorphic and non vanishing on $U_i$?
At first glance these two question looks quite different, but the book says this is the generalization of the Weierstrass problem. How can I see this?
Let $F \subset \Omega$ be a closed nowhere dense subset (where $\Omega$ is a domain of $\mathbb{C}$). Consider the associated Weierstrass problem with multiplicity function $m: F \rightarrow \mathbb{N}$.
Define the following instance of the second Cousin problem: the open subsets are $U_t$, $t \in F$, and $U_{\infty}$, by:
For $t \in F$, $U_t \subset \Omega$ is an open subset such that $U_t \cap F=\{t\}$, and $f_t(z)=(z-t)^{m(t)}$.
Let $U_{\infty}=\Omega \backslash F$ and $f_{\infty}=1$.
Then solving this instance of the second Cousin problem is equivalent to solving the original Weierstrass problem!
More conceptually (or geometrically), the second Cousin problem asks whether effective Cartier divisors are principal – but in dimension $1$, it’s easy to see that effective Cartier divisors correspond exactly to “multiplicity functions” (such as above).