Evaluation of the infinite Dedekind-eta products defining the two-dimensional zeta function

Develop a systematic truncation and numerical evaluation method for the infinite product over prime orbits of Dedekind eta functions that defines the two-dimensional spatiotemporal zeta function 1/ζ[β,z].

Background

In two dimensions, each prime-orbit contribution to the zeta function is an Euler function that can be written in terms of the Dedekind eta function with a complex phase related to the orbit’s stability exponent. The full zeta is an infinite product over such contributions.

While modular-function representations are available, there is currently no general approach to systematically truncate and numerically evaluate the resulting infinite products across many prime orbits, which is needed for practical computations of observables.