Zeta function regularization of high temperature expansions in field theory

Exact high-temperature expansions of the thermodynamic potentials Ωsub(B,F)(β,M,μ) for the relativistic Bose and Fermi gases with temperature T=1/β, mass M and chemical potential μ are derived. (Calculations are done for arbitrary space-time dimension.) The calculations encounter various divergent series which are easily regulated using the Riemann epsilon-function and related functions. The high-T expansions for the Fermi gas are given for the first time. Our results for the Bose gas agree exactly with the high-T series for Ωsub(B) obtained recently by Haber and Weldon (except for a one-term discrepancy arising from the branch points in this fuction). As will be reported more fully elsewhere, out method also solves the more general problem of calculating the high-T, series for the (one-loop) thermodynamic potentials Ω(β,M,μ,Asub(o)) encountered in gauge theories, which depend on additional vacuum parameters (i.e. the diagonal elements of the constant euclidean gauge potential (A0,0)). Beyond this, innumerable mathematical identities of a particular type (useful in contexts other than temperature field theory) can easily be obtained by the epsilon-function method. (orig.)