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Power Partitions and Hayman Functions

This paper establishes that the generating functions for partitions into $k$ -th powers and distinct $k$ -th powers are Hayman functions within the Khinchin family framework, thereby providing a direct derivation of the Hardy–Ramanujan asymptotic formula for these partition counts.

Original authors: José L. Fernández, Víctor J. Maciá

Published 2026-06-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: José L. Fernández, Víctor J. Maciá

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Counting Ways to Build Numbers

Imagine you have a giant bag of Lego bricks. Some bricks are size 1, some are size 8 (which is $2^3$ ), some are size 27 ( $3^3$ ), and so on. These are "perfect cubes."

A partition is simply a way to stack these bricks to build a tower of a specific total height (let's say height $n$ ).

If you only have size 1 bricks, there's only one way to build a tower of height 5 (five 1s).
If you have size 1 and size 8 bricks, you can't build a tower of height 5 using the big bricks, so you still just use the small ones.
But if you want to build a tower of height 10, you could use ten 1s, or one 8 and two 1s.

The mathematicians in this paper, José L. Fernández and Víctor J. Macía, are interested in a specific rule: What if we only use bricks that are perfect $k$ -th powers? (e.g., squares, cubes, fourth powers). They want to know: As the tower gets taller and taller (as $n$ goes to infinity), how many different ways can we build it?

For a long time, mathematicians have known the answer to this question. It looks like a specific formula involving an exponential explosion (the number of ways grows incredibly fast). However, the old ways of proving this formula were like solving a complex maze using a sledgehammer—very heavy, complicated, and involving difficult techniques like the "circle method" or "saddle-point method."

The New Approach: The "Probabilistic" Lens

The authors of this paper didn't just want to re-derive the answer; they wanted to show why the answer makes sense using a different tool: Probability.

Think of the generating function (a fancy math equation that holds all the counting information) not as a static list of numbers, but as a machine that creates random towers.

The Machine (Khinchin Family): They imagine a machine that randomly picks bricks to build a tower. The machine has a "dial" (a variable $t$ ) that controls how likely it is to pick big bricks versus small bricks.
The Average: As you turn the dial, the average height of the towers the machine builds changes.
The Bell Curve (Gaussianity): The authors prove that if you look at the distribution of tower heights the machine produces, they don't just look random; they look like a perfect Bell Curve (the "Normal Distribution" you see in test scores or heights of people).

They call this property "Strongly Gaussian." It's like saying, "If you run this machine a million times, the results will cluster so perfectly around the average that we can predict the outcome with extreme precision."

The "Hayman" Connection: The Master Key

The paper introduces a concept called a "Hayman function." Think of this as a "Gold Standard" certification for these mathematical machines.

The Certification: If a machine is "Hayman," it guarantees that the Bell Curve behavior is so strong and stable that we can use a specific, pre-made "Master Key" (Hayman's Asymptotic Formula) to instantly calculate the number of ways to build a tower of any height.
The Achievement: The authors prove that the machine for "power partitions" (using $k$ -th powers) and the machine for "distinct power partitions" (where you can't use the same brick size twice) are both Hayman functions.

How They Did It (The "Detective Work")

To prove their machine was a "Hayman function," they had to check two things:

The "Major Arc" (The Center): They had to show that the middle of the Bell Curve is perfectly smooth. They used a tool called the "Fulcrum" (a mathematical lever) to analyze the shape of the curve. They proved that the curve is so smooth that it behaves exactly like a perfect Bell Curve.
The "Minor Arc" (The Edges): They had to show that the tails of the curve (the very rare, extreme outcomes) die out quickly enough. For this, they borrowed a powerful estimate from other mathematicians (Tenenbaum, Wu, and Li). Think of this as using a high-powered telescope to prove that the "noise" at the edges of the data is negligible.

The Result: The Formula Falls Out Naturally

Once they proved the machine was "Strongly Gaussian" and a "Hayman function," the rest was easy. They didn't need to do the heavy lifting of the old methods. They just plugged the "average" and "variance" (how spread out the data is) of their machine into the Master Key formula.

The Result:
The famous formula by Hardy and Ramanujan (which predicts how the number of partitions grows) popped out directly and cleanly.

For general partitions ( $k=1$ ): It matches the classic result.
For cubes, fourth powers, etc. ( $k>1$ ): It confirms the formula works for all these cases, but derived through the lens of probability and randomness rather than complex analysis.

The "Distinct" Twist

In the final section, they looked at a slightly different game: Distinct Partitions. Here, you can't use the same brick size twice (e.g., you can't use two size-8 bricks; you can only use one).

They showed that even with this stricter rule, the machine still behaves like a perfect Bell Curve.
They proved it is also a "Hayman function."
This allowed them to write down the exact formula for counting these distinct partitions, confirming results that were known but hard to derive.

Summary

In simple terms, this paper says:

"We took a very hard counting problem about building towers with specific bricks. Instead of using the usual heavy math tools, we treated the problem like a game of chance. We proved that the 'random tower builder' behaves so perfectly (like a Bell Curve) that we can use a standard 'Master Key' to unlock the answer. This confirms the famous formulas for how fast these numbers grow, but it does so in a way that feels more natural and intuitive."

The paper is dedicated to the memory of Christian Pommerenke, a mathematician who likely appreciated this elegant, probabilistic approach to a classic problem.

Technical Summary: Power Partitions and Hayman Functions

Problem Statement
The paper addresses the asymptotic behavior of the number of partitions of an integer $n$ into perfect $k$ -th powers, denoted $p_k(n)$ . While the asymptotic formula for $p_k(n)$ was previously established by Wright, Vaughan, Gafni, and Tenenbaum, Wu, and Li using methods such as the circle method and the saddle-point method, this work seeks to re-derive the result within a specific probabilistic framework. The authors focus on the generating function $P_k(z) = \prod_{n=1}^\infty (1-z^{n^k})^{-1}$ and its counterpart for distinct $k$ -th powers, $Q_k(z) = \prod_{n=1}^\infty (1+z^{n^k})$ . The goal is to demonstrate that these generating functions belong to the "Hayman class" and are "strongly Gaussian," thereby allowing the derivation of the Hardy–Ramanujan asymptotic formula via general theorems on strongly Gaussian power series.

Methodology
The authors employ the theory of Khinchin families, a probabilistic framework associated with power series having non-negative coefficients. For a generating function $f(z)$ , a Khinchin family $(X_t)$ is defined on the non-negative integers with probabilities proportional to the coefficients of $f(z)$ scaled by $t^n$ .

The core methodology involves three main steps:

Probabilistic Characterization: The authors analyze the characteristic function of the normalized random variable $\breve{X}_t = (X_t - m_f(t))/\sigma_f(t)$ , where $m_f(t)$ and $\sigma_f(t)$ are the mean and variance of the Khinchin family. They utilize the concept of the fulcrum $F(z) = \ln(f(e^z))$ to relate the analytic properties of the generating function to the probabilistic moments of the family.
Verification of Strong Gaussianity: To apply Hayman's asymptotic formula, the generating function must be "strongly Gaussian." This requires proving that the variance $\sigma_f(t) \to \infty$ $σ_{f} (t) \to \infty$ as $t$ $t$ approaches the radius of convergence, and that the characteristic function converges to the standard normal characteristic function $e^{-\theta^2/2}$ $e^{- θ^{2} /2}$ in an $L^1$ $L^{1}$ sense over the relevant interval.
- The authors establish Gaussianity using a criterion involving the derivatives of the fulcrum (Theorem A).
- To prove the stronger condition, they rely on analytic bounds for the generating function derived by Tenenbaum, Wu, and Li (specifically Lemma B and Lemma C in the paper). These bounds control the decay of the generating function away from the real axis, which is crucial for bounding the characteristic function in the "minor arc" regions.
Asymptotic Computation: Once strong Gaussianity is established, the authors apply Hayman's asymptotic formula (Theorem B) and the B´aez-Duarte substitution (Theorem C). This allows them to compute the asymptotic behavior of the coefficients $p_k(n)$ by approximating the mean $m_f(t)$ and variance $\sigma_f(t)$ and solving for the parameter $t_n$ such that $m_f(t_n) = n$ .

Key Contributions and Results

Strong Gaussianity of $P_k$ : The primary result (Theorem 1) proves that the Khinchin family associated with $P_k(z)$ is strongly Gaussian. This is achieved by combining a Gaussianity criterion for Khinchin families with the specific bounds on $P_k$ provided by Tenenbaum, Wu, and Li.
Derivation of the Hardy–Ramanujan Formula: As a direct consequence of strong Gaussianity, the authors derive the asymptotic formula for $p_k(n)$ :
$p_k(n) \sim \frac{\alpha_k}{n^{(3k+1)/(2k+2)}} \exp\left(\beta_k n^{1/(k+1)}\right)$
where $\alpha_k$ and $\beta_k$ are explicit constants depending on $k$ . This derivation (Theorem D) avoids the complex arguments of the circle method or saddle-point method, instead relying on the probabilistic mechanism of verifying strong Gaussianity and computing moments.
Hayman Class Membership: The paper proves (Theorem 2) that $P_k(z)$ belongs to the Hayman class. This is a stronger structural property than strong Gaussianity, defined by specific conditions on the "major arc" and "minor arc" of the characteristic function. The proof involves constructing a specific "cut" function $h(t)$ and verifying the decay conditions using the third derivative of the fulcrum and the bounds from Tenenbaum, Wu, and Li.
Extension to Distinct Partitions: The authors extend these results to the generating function $Q_k(z)$ for partitions into distinct $k$ -th powers. They show that $Q_k$ is also in the Hayman class and is strongly Gaussian, deriving the corresponding asymptotic formula for $q_k(n)$ (Equation 43). This required handling the fact that the coefficients of the logarithm of $Q_k$ are not strictly non-negative for $k \ge 2$ , necessitating a more robust Gaussianity criterion than that used for $P_k$ .

Significance and Scope
The paper claims significance primarily in its methodological approach. The authors state that the asymptotic formula (HR) and full asymptotic expansions were already known through the works of Wright, Gafni, and Tenenbaum, Wu, and Li. Therefore, the contribution of this paper is not the discovery of new asymptotic expansions, but rather the demonstration that these results can be obtained "directly" within the probabilistic framework of Khinchin families.

The authors emphasize that their approach "recasts the problem entirely within the probabilistic framework," where the transition from generating function to coefficient asymptotics relies on a "natural probabilistic mechanism"—specifically, the verification of strong Gaussianity and the computation of mean and variance. They explicitly note that while this framework yields the leading asymptotic term (formula HR), it does not extend to full asymptotic expansions, for which the reader is referred to the aforementioned works. The paper is dedicated to the memory of Christian Pommerenke and builds upon the foundational work of Hayman, Rosenbloom, and B´aez-Duarte regarding Khinchin families.