We describe the topology of superlevel sets of (α-stable) Lévy processes X by introducing so-called stochastic ζ-functions, which are defined in terms of the widely used Pers pfunctional in the theory of persistence modules. The latter share many of the properties commonly attributed to ζ-functions in analytic number theory, among others, we show that for α-stable processes, these (tail) ζ-functions always admit a meromorphic extension to the entire complex plane with a single pole at α, of known residue and that the analytic properties of these ζ-functions are related to the asymptotic expansion of a dual variable, which counts the number of variations of X of size ≥ ε. Finally, using these results, we devise a new statistical parameter test using the topology of these superlevel sets.