  We study the one-dimensional Bose-Hubbard model describing the
superfluid-Mott insulator quantum phase transition of cold atoms in optical
lattices. We show that derivatives of the variance of the on-site atom number
occupation, computed with respect to the parameter driving the transition, have
extrema that are located off the critical point even in the thermodynamic
limit. We discuss whether such extrema provide solid evidence of the quantum
Berezinskii-Kosterlitz-Thouless transition taking place in this system. The
calculations are done for systems with the mean number of atoms per lattice
site equal to either one or two. They also characterize the nearest-neighbor
correlation function, which is typically discussed in the context of
time-of-flight images of cold atoms.
