The Riemann zeros as spectrum and the Riemann hypothesis

Abstract

We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros.

33 pages, 17 figures, new abstract, simplification of several sections, changes of references

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• The Generalised Born Oscillator and the Berry-Keating Hamiltonian
• Randomness of Mobius coefficents and brownian motion: growth of the Mertens function and the Riemann Hypothesis
• Riemann zeros as quantized energies of scattering with impurities
• Functional equations for regularized zeta-functions and diffusion processes
• Hamiltonian for the Hilbert-Pólya Conjecture
• Phenomenological formula for Quantum Hall resistivity based on the Riemann zeta function
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry