  Quantum gravity is expected to gauge all global symmetries of effective
theories, in the ultraviolet. Inspired by this expectation, we explore the
consequences of gauging CPT as a quantum boundary condition in phase space. We
find that it provides for a natural semiclassical regularisation and
discretisation of the continuous spectrum of a quantum Hamiltonian related to
the Dilation operator. We observe that the said spectrum is in correspondence
with the zeros of the Riemann zeta and Dirichlet beta functions. Following
ideas of Berry and Keating, this may help the pursuit of the Riemann
hypothesis. It strengthens the proposal that this quantum Hamiltonian captures
the near horizon dynamics of the scattering matrix of the Schwarzschild black
hole, given the rich chaotic spectrum upon discretisation. It also explains why
the spectrum appears to be erratic despite the unitarity of the scattering
matrix.
