Power partitions and Khinchin families

TL;DR

This work develops a probabilistic proof of the Hardy–Ramanujan-type asymptotics for the number $p_k(n)$ of partitions of $n$ into $k$-th powers using Khinchin families. By establishing Gaussianity and strong Gaussianity for the Khinchin family associated to the power-partition generating function $P_k(z)=\prod_{n\ge1}(1-z^{n^k})^{-1}$ and deriving precise asymptotics for the mean and variance, the authors apply Hayman’s framework with Baéz-Duarte substitution to extract explicit constants. The main result is $p_k(n) \sim \alpha_k n^{-(3k+1)/(2k+2)} e^{\beta_k n^{1/(k+1)}}$ with $\beta_k=(k+1)\Omega_k^{k/(k+1)}$ and $\alpha_k=\frac{\Omega_k^{k/(k+1)}}{(2\pi)^{(k+1)/2}\sqrt{1+1/k}}$, where $\Omega_k=\frac{1}{k}\zeta(1+1/k)\Gamma(1+1/k)$. Extensions include partitions into distinct $k$-th powers with analogous asymptotics, connected to prior circle-method results. The approach highlights a concise probabilistic route to HR-type formulas via strong Gaussianity and substitution.

Abstract

We prove, 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: \[ p_k(n) \sim \frac{α_k}{n^{(3k+1)/(2k+2)}} \exp\!\bigl(β_k\, n^{1/(k+1)}\bigr), \quad n \to \infty, \] where $α_k$ and $β_k$ are explicit constants depending only on $k$. The argument reduces to verifying strong Gaussianity of the associated Khinchin family and computing \newline asymptotic approximations of its mean and variance.

Theorems & Definitions (23)

• Theorem A
• Theorem B: Hayman's asymptotic formula
• Theorem C: Báez-Duarte substitution
• Lemma A
• proof : Sketch of proof
• Corollary A
• proof
• Lemma 2.1
• Corollary 2.1
• proof : Proof of Lemma \ref{['lemma: asymptotic_log_m_2ndorder']}