Polylogarithms and Riemann's zeta function

Riemann's zeta 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 zeta 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 zeta 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 zeta 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.