Conventional zeta-function derivation of high-temperature expansions

Zeta-function regularisation (of functional determinants) is used to derive the exact high-temperature expansions of the thermodynamic potentials for the ideal massive Bose and Fermi gases for non-zero chemical potential. The purpose is to show agreement with the results obtained by the Mellin transform method, and by the method of zeta-function regularisation of infinite series. The generalised zeta functions calculated are the simplest examples of Feynman diagram zeta functions. The extension of these calculations to Euclidean manifolds M*En more complicated than the finite-temperature cylinder S1*En is discussed.