  We use Bogoliubov theory to calculate the beyond mean field correction to the
equation of state of a weakly interacting Bose gas in the presence of a tight
2D optical lattice. We show that the lattice induces a characteristic 3D to 1D
crossover in the behaviour of quantum fluctuations. Using the hydrodynamic
theory of superfluids, we calculate the corresponding shift of the collective
frequencies of a harmonically trapped gas. We find that this correction can be
of the order of a few percent and hence easily measurable in current
experiments. The behavior of the quantum depletion of the condensate is also
discussed.
