Black hole normal modes have intriguing connections to logarithmic spectra, and the spectral form factor (SFF) of E_n = log n is the mod square of the Riemann zeta function (RZF). In this paper, we first provide an analytic understanding of the dip-ramp-plateau structure of RZF and show that the ramp at \beta =\Re(s)=0 has a slope precisely equal to 1. The s=1 pole of RZF can be viewed as due to a Hagedorn transition in this setting, and Riemann's analytic continuation to Re(s)< 1 provides the quantum contribution to the truncated log n partition function. This perspective yields a precise definition of RZF 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 O(1)--remarkably, the RZF also exhibits this behavior. To our knowledge, this is the first black hole-inspired toy model that has a demonstrably O(1) Thouless time. In contrast, it is O(log N) in the SYK model and expected to be O(N^{#}) in supergravity fuzzballs. We trace the origins of the ramp to a certain reflection property of the functional equation satisfied by RZF, 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.


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