  Rapidly convergent expansions of a one-loop contribution to the partition
function of quantum fields with ellipsoid constant-energy surface dispersion
law are derived. The omega-potential is naturally decomposed into three parts:
the quasiclassical contribution, the contribution from the branch cut of the
dispersion law, and the oscillating part. The low- and high-temperature
expansions of the quasiclassical part are obtained. An explicit expression and
a relation of the contribution from the cut with the Casimir term and vacuum
energy are established. The oscillating part is represented in the form of the
Chowla-Selberg expansion for the Epstein zeta function. Various resummations of
this expansion are considered. The developed general procedure is applied to
two models: massless particles in a box both at zero and non-zero chemical
potential; electrons in a thin metal film. The rapidly convergent expansions of
the partition function and average particle number are obtained for these
models. In particular, the oscillations of the chemical potential of conduction
electrons in graphene and a thin metal film due to a variation of sizes of the
crystal are described.
