The partition function for a quantum mechanical system can - at least formally - be expressed as a high-temperature series in terms of the \(\zeta\)-function of the system’s Hamiltonian operator. This is done explicitly for a system of noninteracting harmonic oscillators. The Hamiltonian \(\zeta\)-function for the oscillators belongs to a class of \(\zeta\)-functions \(\sum f(n_ i)(n_ 1\omega_ 1+n_ 2\omega_ 2+...+n_ N\omega_ N+a)^{-s}\) constructed from eigenvalues linear in the nonnegative summation indices \(n_ i=0,1,2,... \). Very little is known about such \(\zeta\)-functions. An investigation of particular \(\zeta\)-functions of this type is presented. Much more general results of this class of \(\zeta\)-function have been obtained by the author, and will be presented elsewhere.