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We construct a Hamiltonian H whose discrete spectrum contains, in a certainlimit, the Riemann zeros. H is derived from the action of a massless Diracfermion living in a domain of Rindler spacetime, in 1+1 dimensions, that has aboundary given by the world line of a uniformly accelerated observer.  The action contains a sum of delta function potentials that can be viewed aspartially reflecting moving mirrors. An appropriate choice of the accelerationsof the mirrors, provide primitive periodic orbits associated to the primenumbers $p$, whose periods, measured by the observer's clock, are log p. Actingon the chiral components of the fermion, H becomes the Berry-KeatingHamiltonian (x p + p x)/2, where x is identified with the Rindler spatialcoordinate and p with the conjugate momentum.  The delta function potentials give the matching conditions of the fermionwave functions on both sides of the mirrors. There is also a phase shift forthe reflection of the fermions at the boundary where the observer sits. Theeigenvalue problem is solved by transfer matrix methods in the limit where thereflection amplitudes become infinitesimally small. We find that for genericvalues of the phase shift the spectrum is a continuum, where the Riemann zerosare missing, as in the adelic Connes model. However, for some values of phaseshift, related to the phase of the zeta function, the Riemann zeros appear asdiscrete eigenvalues immersed in the continuum.  We generalize this result to the zeros of Dirichlet L-functions, associatedto primitive characters, that are encoded in the reflection coefficients of themirrors. Finally, we show that the Hamiltonian associated to the Riemann zerosbelongs to class AIII, or chiral GUE, of Random Matrix Theory.

The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime