We elaborate on an entry of the AdS/CFT dictionary relating functional determinants: the determinant of the one-loop contribution to the effective gravitational action by bulk scalars in an asymptotically locally AdS background X, and the determinant of the two-point function of the dual operator (a.k.a. scattering matrix) at the conformal boundary M. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk gravitational partition function to a subleading large-N contribution in the boundary CFT partition function: The formula applies to quotients of AdS as well . In the particular case of the BTZ black hole, a closed expression can be worked out in terms of an associated Patterson-Selberg zeta function ZBTZ (λ) . The determinants can then be thought of as regularized products of either zeta zeros, scattering resonances or quasinormal frequencies . In this sense, one could say that the zeros of ZBTZ (λ) can be heard in the spectrum of quasinormal modes of the BTZ black hole. The question we want to pose is whether a similar situation might exist for the celebrated zeros of the Riemann zeta function.
ASJC Scopus subject areas
- General Physics and Astronomy