Power Partitions and Hayman Functions

Abstract

We prove, within the probabilistic framework of Khinchin families, that the generating function Pk of partitions into k-th powers is strongly Gaussian in the sense of Báez-Duarte, and even further that it is a Hayman function. Thus the Hardy--Ramanujan asymptotic formula for the number pk(n) of partitions of n into k-th powers which reads \[ pk(n) αkn(3k+1)/(2k+2) \!(βk\, n1/(k+1)), n∞, \] where αk and~βk are explicit constants depending only on k, follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity of Pk combines a Gaussianity criterion for Khinchin families with certain bounds of Tenenbaum, Wu and Li on the generating function; the asymptotic formula is recovered by computing asymptotic approximations of the mean and variance of the associated family. Analogous results are presented for the generating function Qk of partitions into distinct k-th powers.