The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime (Q2878630)

A method to the spectral realization of the Riemann zeros is presented. The basic idea is based on the construction of a Hamiltonian whose discrete spectrum contains, in certain limit, the Riemann zeros. It is derived from the action of a massless Dirac fermion in a domain of Rindler space-time, the latter being the geometry associated to accelerated moving observers. The action contains a sum of delta function potentials associated to the prime numbers. The reason is that the properties of the accelerated objects can be used to encode and process arithmetic information. The respective potentials are viewed as a set of partially reflecting accelerated mirrors. An ideal array of such mirrors is constructed and the reflections of the light rays emitted and absorbed by an accelerated observer are studied. The matching conditions for the fermion wave function are then derived. It is shown that the corresponding Hamiltonian is self-adjoint for generic values of the accelerations and reflection coefficients. The eigenvalue problem for the Hamiltonian is solved by the transfer matrix methods. The conditions for the existence of discrete eigenvalues are established in a semi-classical limit. It is proven that for some values of the phase shift, related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues immersed in a continuum spectrum of the Hamiltonian. Finally, it is suggested the possibility of an experimental observation of the Riemann zeros as energy levels of the Hamiltonian using cold atoms, optical lattices or quantum Hall effect. Also, it is pointed down the relation between Riemann zeros and black holes, whose near horizon geometry is that of a Rindler space-time.