Power partitions and Khinchin families

We prove that the generating function of partitions into $k$-th powers is strongly Gaussian in the sense of Báez-Duarte. Within the probabilistic framework of Khinchin families, the Hardy–Ramanujan asymptotic formula for the number~$p_k(n)$ of partitions of~$n$ into $k$-th powers reads $$p_k(n)\sim \frac{{\alpha }_k}{n^{(3k+1)\mathrm{/}(2k+2)}}\exp \left({\beta }_k{n}^{1\mathrm{/}(k+1)}\right),n\to \mathrm{\infty },$$ where ${\alpha }_k$ and ${\beta }_k$ are explicit constants depending only on~$k$, then follows directly from Hayman’s asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity combines a Gaussianity criterion for Khinchin families with 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.