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Berry and Keating conjectured that the classical Hamiltonian H = xp isrelated to the Riemann zeros. A regularization of this model yieldssemiclassical energies that behave, in average, as the non trivial zeros of theRiemann zeta function. However, the classical trajectories are not closed,rendering the model incomplete. In this paper, we show that the Hamiltonian H =x (p + l_p^2/p) contains closed periodic orbits, and that its spectrumcoincides with the average Riemann zeros. This result is generalized toDirichlet L-functions using different self-adjoint extensions of H. We discussthe relation of our work to Polya's fake zeta function and suggest anexperimental realization in terms of the Landau model.

The H=xp model revisited and the Riemann zeros