  We study numerically the low temperature behavior of a one-dimensional Bose
gas trapped in an optical lattice. For a sufficient number of particles and
weak repulsive interactions, we find a clear regime of temperatures where
density fluctuations are negligible but phase fluctuations are considerable,
i.e., a quasicondensate. In the weakly interacting limit, our results are in
very good agreement with those obtained using a mean-field approximation. In
coupling regimes beyond the validity of mean-field approaches, a
phase-fluctuating condensate also appears, but the phase-correlation properties
are qualitatively different. It is shown that quantum depletion plays an
important role.
