Black hole normal modes have intriguing connections to logarithmic spectra, and the spectral form factor (SFF) of En = log n is the mod square of the Riemann zeta function ζ(s). In this paper, we first provide an analytic understanding of the dip-ramp-plateau structure of ζ(s) and show that the ramp at \( \beta \equiv \mathfrak{R}(s)=0 \) has a slope precisely equal to 1. The s = 1 pole of ζ(s) can be viewed as due to a Hagedorn transition in this setting, and Riemann’s analytic continuation to \( \mathfrak{R}(s)<1 \) provides the quantum contribution to the truncated log n partition function. This perspective yields a precise definition of ζ(s) as the “full ramp after removal of the dip”, and allows an unambiguous determination of the Thouless time. For black hole microstates, the Thouless time is expected to be \( \mathcal{O}(1) \) — remarkably, the ζ(s) also exhibits this behavior. To our knowledge, this is the first black hole-inspired toy model that has a demonstrably \( \mathcal{O}(1) \) Thouless time. In contrast, it is \( \mathcal{O}\left(\log N\right) \) in the SYK model and expected to be \( \mathcal{O}\left({N}^{\#}\right) \) in supergravity fuzzballs. We trace the origins of the ramp to a certain reflection property of the functional equation satisfied by ζ(s), and suggest that it is a general feature of L-functions — we find evidence for ramps in large classes of L-functions. As an aside, we also provide an analytic determination of the slopes of (non-linear) ramps that arise in power law spectra using Poisson resummation techniques.
