The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2

This paper proves the convergence of the ratio between Apéry-like and Domb numbers to a specific multiple of $\zeta(3)$ and establishes a related infinite series identity, thereby confirming the Ramanujan Machine conjecture for $Z_2$ through the application of modular forms and eta products.

The Gist

Imagine you are trying to solve a massive, infinite puzzle. The pieces of this puzzle are numbers, and they follow a very specific, rhythmic pattern. This paper is about finding the "secret key" that unlocks the final shape of this puzzle, a key that turns out to be connected to one of the most famous constants in mathematics: $\zeta(3)$ (pronounced "zeta-three").

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Two Rival Teams: The Domb Numbers and Their "Shadow"

The paper starts with two teams of numbers, let's call them Team D and Team B.

• Team D (The Domb Numbers): These are the "stars." They are a sequence of integers that appear in physics (like how particles walk randomly) and pure math. They grow incredibly fast, like a snowball rolling down a mountain.
• Team B (The Companion Sequence): These are the "shadow" team. They follow the exact same rules of movement as Team D, but they start with different initial conditions. They are rational numbers (fractions), not just whole numbers.

The Big Question: If you take a member from Team B and divide it by the corresponding member from Team D, and you keep doing this for larger and larger numbers, what number do you get?

The author proves that this ratio settles down to a specific, beautiful value: $\frac{7}{24} \zeta(3)$.

Think of it like two runners on a track. Runner D is running at a super-fast, predictable speed. Runner B is running alongside them. Even though they start differently, as they run forever, the distance between them (relative to their speed) stabilizes into a perfect, constant ratio. That ratio is the "Apéry-limit."

2. The Ramanujan Machine: A Magical Fraction Machine

The paper also solves a mystery left by the "Ramanujan Machine." Imagine a machine that builds a giant, never-ending fraction (a continued fraction).

• The Machine: It takes a specific formula and spits out a fraction that goes on forever:
  $$2 + \frac{-16 \cdot 16}{36 + \frac{-16 \cdot 26}{160 + \dots}}$$
• The Mystery: Mathematicians guessed that if you let this fraction go on forever, it would equal $\frac{12}{7} \zeta(3)$. But nobody could prove it.
• The Connection: The author shows that this magical fraction machine is actually just a different way of writing the ratio between Team B and Team D. Since we just proved the ratio of the teams is $\frac{7}{24} \zeta(3)$, the fraction machine must equal $\frac{12}{7} \zeta(3)$. The puzzle is solved!

3. The Secret Weapon: The "Modular Map"

How did the author prove this? They didn't just crunch numbers; they used a "map" to translate the problem from the world of simple numbers into a more magical world called Modular Forms.

• The Analogy: Imagine you are trying to understand the shape of a complex 3D object, but you can only see its shadow on a 2D wall. It's hard to tell what the object is.
• The Translation: The author used a special "modular map" (involving things called eta-products and elliptic curves) to lift the problem off the flat wall and into a 3D space where the rules are clearer.
• The "Eichler Integral": In this magical space, the sequence B isn't just a list of numbers; it's a "shadow" of a deeper mathematical object called an Eichler integral. This object behaves like a fluid that flows according to strict laws.

4. The "Atkin-Lehner" Mirror

The author used a special symmetry trick called an Atkin-Lehner transformation.

• The Metaphor: Imagine you have a mirror placed at a specific angle in this magical space. If you look at your reflection, it's not just a copy; it's a transformed version of you.
• The Discovery: The author looked at the "fluid" (the Eichler integral) in the mirror. They found that the fluid in the mirror and the fluid in reality are related by a specific equation involving $\zeta(3)$.
• The Fixed Point: There is a special spot in this space (a "fixed point") where the mirror reflects the object onto itself. By analyzing exactly how the fluid behaves at this specific spot, the author could calculate the exact ratio of the two teams (D and B).

5. The Final Result

By combining the "map" (modular parametrization) with the "mirror" (Atkin-Lehner transformation), the author calculated the exact behavior of the numbers as they go to infinity.

• The Limit: The ratio of the companion sequence to the Domb numbers is exactly $\frac{7}{24} \zeta(3)$.
• The Sum: This also proves a specific infinite sum involving the Domb numbers equals $\frac{56}{3} \zeta(3)$.
• The Conjecture: This confirms the Ramanujan Machine's guess about the continued fraction.

Summary in One Sentence

The author proved that two specific sequences of numbers, which seem unrelated at first, converge to a ratio defined by the famous constant $\zeta(3)$, by translating the problem into a "magical" geometric space where symmetries reveal the hidden answer.

Why does this matter?
It connects three different areas of math:

1. Combinatorics (counting paths and numbers).
2. Number Theory (properties of prime numbers and constants like $\zeta(3)$).
3. Geometry (shapes and symmetries in complex space).

It's like finding out that the pattern of leaves on a tree, the rhythm of a heartbeat, and the shape of a snowflake are all governed by the same underlying musical note.

Technical Summary

1. Problem Statement

The paper addresses three interconnected problems in the theory of special functions, modular forms, and number theory:

1. The Domb Apéry-limit: Determining the exact limit of the ratio between the rational companion sequence $B_n$ and the Domb numbers $D_n$ as $n \to \infty$. The Domb numbers are defined by $D_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{2k}{k} \binom{2(n-k)}{n-k}$. While the limit was listed as a known value ($\frac{7}{24}\zeta(3)$) in previous literature (Almkvist et al.), a rigorous proof was missing.
2. The Domb Sum Identity: Proving the experimental identity $\sum_{n \ge 1} \frac{64n}{n^3 D_n D_{n-1}} = \frac{56}{3}\zeta(3)$, which appeared in Cohen's work on the Ramanujan Machine.
3. Ramanujan Machine Conjecture Z2: Proving that a specific continued fraction, generated by the Ramanujan Machine algorithm, converges to $\frac{12}{7}\zeta(3)$. The continued fraction is defined by the partial numerators and denominators derived from the recurrence coefficients of the Domb numbers.

2. Methodology

The author employs a sophisticated blend of modular forms, Eichler integrals, and asymptotic analysis. The proof strategy proceeds in the following logical stages:

A. Recurrence and Discrete Wronskian Analysis

• The Domb numbers $D_n$ and the companion sequence $B_n$ satisfy the same third-order linear recurrence relation.
• Using a discrete Wronskian argument ($W_n = D_n B_{n-1} - D_{n-1} B_n$), the author establishes a telescoping sum identity:
  $$\sum_{n=1}^N \frac{64n}{n^3 D_n D_{n-1}} = 64 \frac{B_N}{D_N}$$
• This reduces the proof of the infinite sum (Theorem 1.2) and the continued fraction limit (Theorem 1.3) directly to the proof of the Apéry-limit (Theorem 1.1).

B. Modular Parametrization

• The generating functions $A(z) = \sum D_n z^n$ and $B(z) = \sum B_n z^n$ are linked to modular forms via a level-6 eta-product parametrization (Chan-Zudilin/Zhou).
• The variable $z$ is identified with a modular function $\xi(\tau) = -\frac{\eta(2\tau)^6 \eta(6\tau)^6}{\eta(\tau)^6 \eta(3\tau)^6}$.
• The companion sequence $B_n$ corresponds to an inhomogeneous solution of the differential equation satisfied by $A(z)$. This allows the definition of a ratio function $E(\tau) = B(\xi(\tau))/A(\tau)$.

C. Eichler Integral and Atkin-Lehner Involution

• The function $E(\tau)$ is identified as a weight -2 Eichler integral associated with a specific modular form $g(\tau)$ of weight 4.
• The form $g(\tau)$ is constructed from Eisenstein series $E_4$ and is shown to be an eigenform under the Atkin-Lehner involution $W = \begin{pmatrix} 3 & -2 \\ 6 & -3 \end{pmatrix}$ with eigenvalue $-1$(i.e.,$g|_4 W = -g$).
• The author utilizes the Mellin transform of $E(\tau)$ to relate its values to the $L$-function of $g$. Specifically, the alternating $L$-series $L^*(s)$ of $g$ is computed, yielding values involving $\zeta(3)$.

D. Local Analysis at the Dominant Singularity

• The asymptotic behavior of $D_n$ and $B_n$ is determined by the singularity of the generating functions at $z_0 = 1/16$.
• This singularity corresponds to an elliptic fixed point $\tau^* = \frac{1}{2} + \frac{i}{2\sqrt{3}}$ of the Atkin-Lehner involution $W$.
• Using the Flajolet-Odlyzko transfer theorem, the ratio of the coefficients $B_n/D_n$ is shown to converge to the ratio of the linear coefficients in the local expansion of $B(\xi(\tau))$ and $A(\tau)$ near $\tau^*$.

E. Period Polynomial Computation

• The core of the proof involves differentiating the transformation law of the Eichler integral under the involution $W$.
• The author proves the identity:
  $$(E|_{-2} W)(\tau) + E(\tau) = \frac{7}{6}\zeta(3)(3\tau^2 - 3\tau + 1)$$
• By differentiating this identity at the fixed point $\tau^*$ and combining it with the transformation law of $A(\tau)$, the author isolates the specific linear combination of $E(\tau^*)$ and its derivative that yields the limit.

3. Key Contributions

1. Rigorous Proof of the Apéry-limit: The paper provides the first complete proof that $\lim_{n \to \infty} \frac{B_n}{D_n} = \frac{7}{24}\zeta(3)$, filling a gap in the literature where this value was previously only conjectured or listed without proof.
2. Proof of the Ramanujan Machine Conjecture Z2: It confirms that the continued fraction generated by the Ramanujan Machine for $\zeta(3)$ converges to $\frac{12}{7}\zeta(3)$. This validates the machine's heuristic algorithm for this specific case.
3. Connection between Combinatorics and Modular Forms: The work demonstrates a deep link between the asymptotic behavior of combinatorial sequences (Domb numbers) and the arithmetic properties of weight-4 modular forms and their associated Eichler integrals.
4. Correction of Naive Expectations: The paper clarifies that the limit is not simply the value of the Eichler integral at the CM point ($E(\tau^*)$), but rather a specific linear combination involving the derivative, correcting a potential misconception about "naive" CM values.

4. Main Results

• Theorem 1.1: $\lim_{n \to \infty} \frac{B_n}{D_n} = \frac{7}{24}\zeta(3)$.
• Theorem 1.2: $\sum_{n \ge 1} \frac{64n}{n^3 D_n D_{n-1}} = \frac{56}{3}\zeta(3)$.
• Theorem 1.3: The continued fraction $2 + \frac{-16 \cdot 1^6}{36 + \frac{-16 \cdot 2^6}{160 + \dots}}$ converges to $\frac{12}{7}\zeta(3)$.

5. Significance

This paper is significant for several reasons:

• Validation of AI-Driven Conjectures: It provides a rigorous mathematical foundation for a conjecture generated by the "Ramanujan Machine," an AI project designed to discover new identities for mathematical constants. This strengthens the credibility of using computational heuristics to guide high-level number theory research.
• New Techniques for Apéry-like Limits: The method of using Atkin-Lehner involutions on Eichler integrals to compute Apéry limits offers a powerful new tool for analyzing other sequences that satisfy similar recurrence relations but lack known closed-form limits.
• Explicit Values for $\zeta(3)$: The results contribute to the growing body of knowledge regarding representations of Apéry's constant $\zeta(3)$ through combinatorial sums and continued fractions, which are central to the study of irrationality measures and transcendental number theory.

In summary, Shvets successfully bridges the gap between experimental mathematics (Ramanujan Machine) and rigorous analytic number theory, proving that the Domb numbers encode deep modular arithmetic properties related to $\zeta(3)$.