  We investigate quantum fluctuations of a vortex lattice in a one-dimensional
optical lattice. Our method gives full access to all the modes of the vortex
lattice and we discuss in particular the Bloch bands of the Tkachenko modes.
Because of the small number of particles in the pancake Bose-Einstein
condensates at every site of the optical lattice, finite-size effects become
very important. Therefore, the fluctuations in the vortex positions are
inhomogeneous and the melting of the lattice occurs from the outside inwards.
Tunneling between neighbouring pancakes substantially reduces the inhomogeneity
as well as the size of the fluctuations.
