幂次分拆与Hayman函数
Power Partitions and Hayman Functions
AI总结：
在Khinchin族的概率框架下，证明分拆成k次幂的生成函数是强高斯的（Hayman函数），从而直接由Hayman渐近公式得到Hardy-Ramanujan渐近公式。
摘要：
我们在Khinchin族的概率框架内证明，分拆成$k$次幂的生成函数$P_k$在Báez-Duarte意义下是强高斯的，甚至更进一步，它是一个Hayman函数。因此，关于$n$分拆成$k$次幂的个数$p_k(n)$的Hardy--Ramanujan渐近公式\[ p_k(n) \sim \frac{\alpha_k}{n^{(3k+1)/(2k+2)}} \exp\!\Big(\beta_k\, n^{1/(k+1)}\Big), \qquad n\to\infty, \]其中$\alpha_k$和$\beta_k$是仅依赖于$k$的显式常数，直接由Hayman关于强高斯幂级数的渐近公式得出。$P_k$的强高斯性的证明结合了Khinchin族的高斯性准则与Tenenbaum、Wu和Li关于生成函数的某些界；通过计算相关族的均值和方差的渐近近似，恢复了渐近公式。对于分拆成不同$k$次幂的生成函数$Q_k，给出了类似的结果。
英文摘要：
We prove, within the probabilistic framework of Khinchin families, that the generating function $P_k$ 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 $p_k(n)$ of partitions of $n$ into $k$-th powers which reads \[ p_k(n) \sim \frac{α_k}{n^{(3k+1)/(2k+2)}} \exp\!\Big(β_k\, n^{1/(k+1)}\Big), \qquad n\to\infty, \] where $α_k$ and~$β_k$ 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 $P_k$ 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 $Q_k$ of partitions into distinct $k$-th powers.
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