  Berry and Keating conjectured that the classical Hamiltonian H = xp is
related to the Riemann zeros. A regularization of this model yields
semiclassical energies that behave, in average, as the non trivial zeros of the
Riemann 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 spectrum
coincides with the average Riemann zeros. This result is generalized to
Dirichlet L-functions using different self-adjoint extensions of H. We discuss
the relation of our work to Polya's fake zeta function and suggest an
experimental realization in terms of the Landau model.
