Convergence of the Schottky–Klein prime-function infinite product for general circular domains

Establish a complete convergence theory for the infinite-product representation of the Schottky–Klein prime function for every multiply connected circular domain, without imposing classical geometric restrictions on the associated Schottky group.

Background

The Schottky–Klein prime function is represented by an infinite product over non-identity elements of the Schottky group associated with a multiply connected circular domain. The paper explains that classical convergence results typically require geometric assumptions, such as circle-decomposability, while general circular-hole configurations may not satisfy these restrictions.

The paper develops a potential-based truncation method and proves an error estimate under the condition that the Hausdorff dimension of the Schottky-group limit set is less than one. It also identifies a graph-based sufficient condition for convergence. These results do not establish convergence for all multiply connected circular domains, leaving the general convergence problem unresolved.