Analytic continuation of the topo-symmetric extension zeta function

Prove that the zeta function ζ_ts(s; G, H) = Σ_{E ∈ Ext_ts(G,H)} ord_max(E)^{-s} associated with topo-symmetric extensions of a topological group G by a discrete abelian group H admits an analytic continuation to the complex plane, and characterize the poles in terms of the arithmetic structure of G and H.

Background

To paper counting and asymptotic properties of topo-symmetric extensions, the authors define a zeta function ζ_ts(s;G,H) that sums over extensions weighted by the reciprocal of their maximal element order. They conjecture that this function possesses analytic continuation and that its poles encode arithmetic information about G and H.

This conjecture parallels classical results for number-theoretic zeta functions and aims to develop analytic tools for understanding the distribution and structure of topo-symmetric extensions.