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Asymptotics of the d'Arcais Numbers at Small $k$

This paper investigates the asymptotic behavior of d'Arcais numbers for fixed small $k$ as $n$ grows large, deriving a formula related to Ramanujan's work, disproving a specific conjecture by Heim and Neuhauser for $k=2$ while confirming it for $k \ge 3$ under large $n$ , and employing the Hardy-Ramanujan circle method as a heuristic tool.

Original authors: Shannon Starr

Published 2026-02-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Shannon Starr

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 Patterns in a Cosmic Library

Imagine you have a giant, infinite library. Inside this library, there are books that represent numbers. But these aren't just ordinary numbers; they are "d'Arcais numbers." Think of them as a special way of organizing or counting how many different ways you can arrange items in a specific pattern.

The author, Shannon Starr, is trying to answer a very specific question: If you pick a specific pattern (let's call it "k") and look at a very large number ("n"), how big is the count of these patterns?

In math terms, this is about finding a "formula" that predicts the size of these numbers when they get huge.

The Main Discovery: A Recipe for Prediction

The paper finds a "recipe" (a mathematical formula) to predict these numbers when the pattern size ( $k$ ) stays small but the total number ( $n$ ) gets massive.

Here is the recipe in plain English:

The Ingredients: The formula uses two main things:
- Divisors: Think of a number like 12. Its divisors are 1, 2, 3, 4, 6, and 12. The formula adds up powers of these divisors.
- Famous Constants: It uses numbers that mathematicians love, like $\zeta(2)$ (which is related to the sum of $1/1^2 + 1/2^2 + \dots$ ).
The Result: The paper proves that for large $n$ , the count of these patterns behaves almost exactly like a specific combination of these divisors and constants.

The Analogy: Imagine you are trying to guess how many grains of sand are on a beach. You can't count them one by one. Instead, you look at a small sample, measure the density, and use a formula to estimate the total. This paper provides that formula for a very specific, complex type of "sand grain" (the d'Arcais numbers).

The "Log-Concavity" Mystery: Is the Hill Smooth?

The paper tackles a famous puzzle proposed by Heim and Neuhauser. They asked: Is the sequence of these numbers shaped like a smooth hill?

The Hill Analogy: Imagine a hill where the height represents the number of patterns.
- If you go up the hill, reach a peak, and come down, the slope gets steeper as you go up and less steep as you go down.
- Mathematically, this is called "log-concavity." It means the numbers don't have weird, jagged spikes or dips; they follow a smooth, predictable curve.

What the Paper Found:

For small patterns ( $k=2$ ): The hill is not smooth. It has a jagged edge. The conjecture that it would be smooth is false for this specific case.
For larger patterns ( $k=3, 4, \dots$ ): Once you get past the first few steps, the hill is smooth. The numbers settle into a predictable, smooth shape when the total number ( $n$ ) is large enough.

How They Solved It: The "Circle Method"

To find these answers, the author used a tool called the Hardy-Ramanujan Circle Method.

The Metaphor:
Imagine you are trying to hear a specific whisper in a noisy stadium.

The Noise: The math is full of complex, swirling numbers.
The Whisper: The answer you are looking for.
The Circle Method: This is like putting on noise-canceling headphones that only let you hear the frequencies coming from specific spots (called "Farey fractions").
The Trick: The author realized that for this specific problem, you don't need to listen to the entire stadium. You only need to focus on the "major arc" (the loudest, most important part of the noise near the real numbers). By ignoring the rest, the math becomes much simpler and easier to solve.

Why This Matters (According to the Paper)

It connects old and new: The paper connects a formula Ramanujan (a legendary mathematician) wrote a century ago with modern research. It shows that Ramanujan's intuition was correct, even if he didn't have the full proof for every detail.
It settles a debate: It proves that the "smooth hill" idea (log-concavity) works for almost all cases, except for the very first, simplest case ( $k=2$ ).
It simplifies the math: Previous attempts to solve this required incredibly complex, multi-layered math. This paper shows you can get the same result using a simpler, single-step approach.

Summary in One Sentence

This paper uses a simplified version of a famous mathematical listening technique to prove that while a specific sequence of numbers gets messy at the very beginning, it eventually settles into a perfectly smooth, predictable pattern for all larger cases.

Technical Summary: Asymptotics of the d'Arcais Numbers at Small k

Problem Statement
The paper investigates the asymptotic behavior of the d'Arcais numbers, denoted as $A(2, n, k)$ , which form a triangular array defined by the generating function:
$\sum_{n=0}^{\infty} \sum_{k=0}^{n} A(2, n, k) \frac{x^k z^n}{n!} = ((z; z)_{\infty})^{-x}$
where $(z; z)_{\infty} = \prod_{n=1}^{\infty} (1 - z^n)$ is the infinite q-Pochhammer symbol. These numbers are combinatorial objects related to the abundance index $\sigma_{-1}(n)$ and can be interpreted via the Bell transform or as counting commuting tuples of permutations.

The primary focus is on the regime where $k$ is fixed and $n \to \infty$ . Specifically, the paper addresses the asymptotic ratio of the normalized numbers $\alpha(n, k) = k! A(2, n, k) / n!$ . This investigation is motivated by a conjecture from Heim and Neuhauser regarding the log-concavity of the sequence $k \mapsto A(2, n, k)$ . The conjecture posits that $A(2, n, k)/A(2, n, k-1) \geq A(2, n, k+1)/A(2, n, k)$ . While previous work by the author and Abdesselam established log-concavity in the "typicality regime" ( $k \sim \sqrt{n}$ ) using central limit theorems, the behavior for small fixed $k$ required distinct analysis.

Methodology
The paper employs two complementary approaches to derive and verify the asymptotic formulas:

Heuristic Circle Method: The author utilizes the Hardy-Ramanujan circle method as a heuristic guide. By analyzing the generating function $F(t) = -\ln((e^{-t}; e^{-t})_{\infty})$ and leveraging the modular symmetry of the Dedekind eta function $\eta(\tau)$ , the paper identifies the dominant contributions from "major arcs" near Farey fractions. This heuristic suggests that the asymptotics are governed by Ramanujan sums and divisor functions, leading to an ansatz for $\alpha(n, k)$ .
Rigorous Arithmetic Convolution Analysis: To prove the formulas rigorously without relying on the full Rademacher decomposition, the paper adapts methods from Ramanujan, Ingham, Halberstam, and more recently Oliver, Shrestha, and Thorne. The core of the proof involves analyzing the $k$ -fold arithmetic convolution of the function $\sigma_{-1}(n)$ . The proof relies on:
- Estimating sums of the form $\sum k^{\alpha-1}(n-k)^{\beta-1}$ over arithmetic progressions using Beta integrals.
- Utilizing properties of Dirichlet series and the identity $\sum_{(m,n)=1} m^{-r}n^{-s} = \zeta(r)\zeta(s)/\zeta(r+s)$ .
- Inductive arguments on $k$ to establish the asymptotic equivalence for the $k$ -fold convolution.

Key Results

Asymptotic Formula for Fixed $k$ : The paper proves that for fixed $k \geq 2$ , as $n \to \infty$ :
$\alpha(n, k) \sim \frac{(\zeta(2))^k}{\Gamma(k)\zeta(2k)} \frac{\sigma_{2k-1}(n)}{n^k} = \frac{(\zeta(2))^k}{\Gamma(k)\zeta(2k)} n^{k-1} \sigma_{-2k+1}(n)$
This result generalizes a formula found by Ramanujan in his paper "On Certain Arithmetical Functions."
Resolution of the Log-Concavity Conjecture: The paper analyzes the ratio $R(n, k) = \frac{(\alpha(n, k))^2}{\alpha(n, k-1)\alpha(n, k+1)}$ to test the Heim-Neuhauser conjecture.
- Case $k=2$ : The conjecture is false. The limit inferior of the ratio is 0:
  $\liminf_{n \to \infty} \frac{(\alpha(n, 2))^2}{\alpha(n, 1)\alpha(n, 3)} = 0$
  This occurs because the term involving $\zeta(2k-3)$ becomes $\zeta(1)$ , which diverges.
- Case $k \geq 3$ : The conjecture is true for sufficiently large $n$ . The limit inferior is strictly greater than 1:
  $\liminf_{n \to \infty} \frac{(\alpha(n, k))^2}{\alpha(n, k-1)\alpha(n, k+1)} = \frac{k}{k-1} \cdot \frac{\zeta(2k+2)\zeta(2k-2)}{(\zeta(2k))^2} \cdot \frac{(\zeta(2k-1))^2}{\zeta(2k-3)\zeta(2k+1)} > 1$
  The proof demonstrates that the sequence is asymptotically log-concave for $k \geq 3$ .

Significance and Claims
The paper claims to provide a "slightly generalized version" of Ramanujan's formula regarding the asymptotics of divisor sums and their convolutions. Its primary contribution is the rigorous derivation of the asymptotic behavior of d'Arcais numbers for small fixed $k$ , a regime where previous large-deviation formulas (valid for $k \to \infty$ ) were insufficient.

The author explicitly states that the results are an "immediate consequence" of the more recent work by Oliver, Shrestha, and Thorne, adapted to the specific context of d'Arcais numbers. The paper does not claim to provide explicit bounds on the size of $n$ required for the asymptotic behavior to hold (unlike the work of Hong and Zhang on Okounkov-Nekrasov polynomials), but rather provides the explicit formula for the limit inferior of the log-concavity ratio.

The work serves to clarify the behavior of the d'Arcais numbers in the "small $k$ " regime, correcting the assumption that the Heim-Neuhauser log-concavity conjecture holds universally for all $k$ , while confirming its validity for $k \geq 3$ in the asymptotic limit. The author notes that extending these asymptotic techniques to $\ell \geq 3$ (the general multi-dimensional case) remains an open problem, as it would likely require bypassing modular forms or utilizing generalized Meinardus theorems.