Evaluation of multidimensional linear zeta-functions (Q915771)

The author considers functions in a complex variable s of the form \(\sum (c_ 1m_ 1+...+c_ Nm_ N)^{-s}\) where \(m_ 1,...,m_ N\) run independently from 1 to infinity and \(c_ 1,...,c_ N\) are real numbers. It is shown by Euler-MacLaurin sum rules and multinomial expansions that such a function can be written as a combination in terms of the Riemann zeta-function, too complicated to be quoted here explicitly. In theoretical physics these functions occur associated with the Hamiltonian operator of a system of N noninteracting harmonic oscillators.