Riemann’s ζ function has been important in statistical mechanics for many years, especially for the understanding of Bose-Einstein condensation. Polylogarithms can yield values of Riemann’s ζ function in a special limit. Recently these polylogarithm functions have unified the statistical mechanics of ideal gases. Our particular concern is obtaining the values of Riemann’s ζ function of negative order suggested by a physical application of polylogs. We find that there is an elementary way of obtaining them, which also provides an insight into the nature of the values of Riemann’s ζ function. It relies on two properties of polylogs—the recurrence and duplication relations. The relevance of the limit process in the statistical thermodynamics is described.

• Received 12 March 1997

DOI:https://doi.org/10.1103/PhysRevE.56.3909