An Analytic Zeta Function Ramp at the Black Hole Thouless Time

TL;DR

This paper analytically explores the spectral form factor of the Riemann zeta function, revealing a black hole-inspired model with a constant Thouless time and connecting ramp phenomena to properties of L-functions.

Contribution

It provides an analytic understanding of the RZF spectral form factor's ramp structure and introduces a black hole microstate model with a constant Thouless time.

Findings

The ramp at Re(s)=0 has a slope exactly equal to 1.

The model exhibits an O(1) Thouless time, unlike other models.

Evidence of ramp phenomena in various L-functions.

Abstract

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 $\beta \equiv \Re(s)=0$ 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 $\mathcal{O}(1)$--remarkably, the RZF also exhibits this behavior. To our knowledge, this is the first black hole-inspired toy model that has a demonstrably $\mathcal{O}(1)$ Thouless time. In contrast, it is $\mathcal{O}(\log N)$ in the SYK model and expected to be $\mathcal{O}(N^{\#})$ 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.