Infinite products, partition functions, and the Meinardus theorem

Abstract

The Meinardus theorem provides an asymptotic value for the infinite-product generating functions of many different partition functions, and thereby asymptotic values for these partition functions themselves. Under assumptions somewhat stronger than those of Meinardus a more specific version of the theorem is derived, then this is generalized from the spectrum of integers considered by Meinardus to infinite products constructed from an arbitrary spectrum. Included are infinite products that generate partition functions able to count states in the quantum theory of extended objects. There emerges a quite general and explicit method (of which only limited details are given here) for deriving the asymptotic growth of the state density function of multidimensional quantum objects. One finds a universal geometry-independent leading behavior, with nonleading geometry-dependent corrections that the method is capable of providing explicitly.