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The level-8 Apery-limit and a proof of the Ramanujan Machine conjecture Z1

This paper provides the first complete proof of the Ramanujan Machine conjecture Z1 by establishing the level-8 Apery-limit as $(7/32)\zeta(3)$ using advanced techniques including eta-product parametrization, Wronskian identities, and Eichler integrals.

Original authors: Alex Shvets

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex Shvets

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

Imagine you are trying to solve a massive, infinite puzzle. On one side of the puzzle, you have a sequence of numbers that grows incredibly fast, following a very specific, complicated rule. On the other side, you have a "companion" sequence that follows the exact same rule but starts with different numbers.

The big question mathematicians have been asking is: As these numbers get infinitely large, what is the ratio between the companion sequence and the main sequence?

This paper, written by Alex Shvets, solves a specific, very hard version of this puzzle (called the "Level-8" problem) and, in doing so, proves a famous guess made by the "Ramanujan Machine" (a project that uses computers to find new mathematical formulas).

Here is the story of how they solved it, broken down into simple concepts.

1. The Two Rival Runners

Think of the two sequences as two runners on a track that stretches to infinity.

Runner A (The $s_n$ sequence): This runner is the "main character." They follow a complex set of instructions (a cubic recurrence relation) to decide their next step.
Runner B (The $B_n$ sequence): This runner follows the exact same instructions but starts from a different spot.

Mathematicians wanted to know: If both runners run forever, how far apart are they? Specifically, if you divide Runner B's distance by Runner A's distance, does it settle on a specific number?

The paper proves that this ratio settles on a very special number: $\frac{7}{32} \times \zeta(3)$ .
(Note: $\zeta(3)$ is a famous constant called Apéry's constant, related to the sum of the cubes of all numbers. It's a bit like $\pi$ , but for cubes.)

2. The Secret Map: The "Modular Parametrization"

To solve this, the author didn't just watch the runners; he looked for a secret map that connects their running path to a different world entirely.

In mathematics, there is a concept called "modular forms." Imagine these as a special kind of musical instrument that plays a song based on the shape of a donut (a torus) in the complex plane.

The author uses a specific "instrument" (an eta-product) that acts like a translator.
This translator converts the complicated running rules of our sequences into a smooth, flowing song (a function of a complex variable $\tau$ ).
Suddenly, the messy, discrete steps of the runners become a smooth, continuous wave. This makes the problem much easier to analyze.

3. The "Eichler Integral" (The Time Machine)

The paper introduces a tool called an Eichler integral. Think of this as a time machine or a shadow projector.

The "main runner" ( $s_n$ ) corresponds to a standard wave.
The "companion runner" ( $B_n$ ) corresponds to the shadow or the integral of that wave.
By studying the shadow, the author can figure out exactly how the companion runner behaves without having to calculate every single step of their run.

4. The Mirror Test (Fricke Involution)

Here is the cleverest part of the proof. The author uses a mathematical "mirror" called the Fricke involution.

Imagine the track has a mirror in the middle. If you look at the runner in the mirror, they look slightly different, but they are still the same person.
The author applies this mirror to the "shadow" (the Eichler integral).
When he adds the original shadow and the mirrored shadow together, something magical happens: All the messy, complicated parts cancel out.
What is left is a simple, clean polynomial (a basic algebraic equation). This polynomial contains the answer we are looking for: the ratio $\frac{7}{32}\zeta(3)$ .

5. The Cliff Edge (Singularity Analysis)

To find the final answer, the author looks at the "edge of the world" for these functions. In math, functions often behave wildly near a specific point called a singularity (like a cliff edge).

The author analyzes how the function behaves as it gets close to this cliff.
He discovers that the function has a "square-root" shape near the edge.
By comparing how the "main runner" and the "companion runner" approach this cliff, he can calculate their final ratio. It's like seeing two cars approach a stop sign; even if they are moving fast, you can predict exactly where they will stop relative to each other based on their speed and angle.

6. The Grand Prize: The Ramanujan Machine Conjecture

The ultimate goal wasn't just to find the ratio; it was to prove a specific Continued Fraction.

A continued fraction is a way of writing a number as a fraction inside a fraction inside a fraction... forever.
The "Ramanujan Machine" (a computer project) had guessed that a specific, very complex continued fraction equals $\frac{8}{7\zeta(3)}$ .
Because the author proved the ratio of the runners is $\frac{7}{32}\zeta(3)$ , the math automatically flips this result to prove the continued fraction is indeed $\frac{8}{7\zeta(3)}$ .

Summary

In short, Alex Shvets took a problem about two sequences of numbers growing infinitely large. He:

Translated them into a smooth "song" using modular forms.
Used a "mirror" trick to simplify the relationship between the two sequences.
Analyzed how they behave near a mathematical "cliff."
Proved that their ratio is exactly $\frac{7}{32}\zeta(3)$ .
Used this to confirm a computer-generated guess about a complex fraction, adding a new piece to the puzzle of mathematical constants.

It's a beautiful example of how connecting different areas of math (number theory, complex analysis, and geometry) can solve problems that seem impossible to solve by brute force.

1. Problem Statement

The paper addresses two interconnected problems in number theory and the theory of special functions:

The Level-8 Apéry-Limit: Determining the asymptotic limit of the ratio between a rational companion sequence $B^{(8)}_n$ and the level-8 Apéry-like sequence $s_n$ as $n \to \infty$ . The sequences are defined by a specific cubic recurrence relation.
The Ramanujan Machine Conjecture Z1: Proving a specific polynomial continued fraction identity conjectured by the "Ramanujan Machine" project. The conjecture states that the continued fraction with partial numerators $a_n = (2n+1)(3n^2+3n+1)$ and partial denominators $b_n = -n^6$ converges to $\frac{8}{7\zeta(3)}$ .

While the limit value $\frac{7}{32}\zeta(3)$ was previously identified numerically and outlined via modularity arguments in an unpublished preprint by Golyshev, a complete, self-contained proof was lacking.

2. Methodology

The author employs a sophisticated blend of modular forms, Eichler integrals, singularity analysis, and continued fraction theory. The proof proceeds through the following logical stages:

A. Modular Parametrization

The paper utilizes the level-8 modular parametrization established by Cooper.

Sequences: The generating functions $A(z) = \sum s_n z^n$ and $B(z) = \sum B^{(8)}_n z^n$ satisfy linear differential equations derived from the recurrence relations.
Eta-Products: The solution is parametrized using modular functions constructed from Dedekind eta functions:
$t(\tau) = \frac{\eta(\tau)^8 \eta(8\tau)^8}{\eta(2\tau)^8 \eta(4\tau)^8}, \quad Y(\tau) = \frac{\eta(2\tau)^6 \eta(4\tau)^6}{\eta(\tau)^4 \eta(8\tau)^4}$
where $Y(\tau) = A(t(\tau))$ .
Fricke Involution: The paper analyzes the action of the Fricke involution $W_8(\tau) = -1/(8\tau)$ on these functions, establishing transformation laws such as $Y(W_8\tau) = -8\tau^2 Y(\tau)$ .

B. Identification as an Eichler Integral

The companion sequence $B_n$ is linked to modular forms via an Eichler integral.

The ratio $E(\tau) = B(t(\tau))/Y(\tau)$ is identified as a normalized weight -2 Eichler integral of a specific weight-4 modular form $g_8(\tau)$ .
The form $g_8$ is explicitly constructed as a linear combination of Eisenstein series $E_4$ :
$g_8(\tau) = \frac{E_4(\tau) - 21E_4(2\tau) + 84E_4(4\tau) - 64E_4(8\tau)}{240}$
Using Sturm's bound, the author proves that the differential operator applied to $E(\tau)$ yields exactly $g_8(\tau)$ .

C. Period Polynomials and Mellin-Barnes Integration

A crucial step involves computing the "period polynomial" associated with the modular form.

The author defines the period polynomial $P_8(\tau) = (E|_{-2}W_8)(\tau) + E(\tau)$ .
By applying the Mellin transform to the modular form $g_8$ and utilizing its functional equation (derived from the properties of the Riemann zeta function), the author computes the residues at critical values.
This calculation yields the explicit polynomial:
$P_8(\tau) = \frac{7}{32}\zeta(3)(8\tau^2 + 1)$
This identity is the core analytic engine of the proof.

D. Singularity Analysis

To extract the asymptotic limit of the coefficients $s_n$ and $B^{(8)}_n$ :

The paper identifies the dominant singularity $t_0$ of the generating function $A(z)$ . This corresponds to the fixed point of the Fricke involution in the upper half-plane, $\tau^* = i/(2\sqrt{2})$ .
The value is found to be $t_0 = \frac{3-2\sqrt{2}}{4}$ .
Using the Flajolet-Odlyzko transfer theorem, the local behavior of $A(z)$ and $B(z)$ near $t_0$ is analyzed. The exponents at the singularity are found to be $0, 1/2, 1$ .
The asymptotic ratio of the coefficients is determined by the ratio of the leading coefficients of the square-root terms in the local expansions of $A(z)$ and $B(z)$ .

E. Continued Fraction Connection

Finally, the paper connects the Apéry limit to the continued fraction.

It is shown that the convergents of the Ramanujan Machine continued fraction are directly proportional to the sequences $s_n$ and $B^{(8)}_n$ (specifically involving factorials and powers of 4).
The limit of the convergents is the reciprocal of the Apéry limit (scaled by a constant factor).

3. Key Contributions

Proof of the Level-8 Apéry-Limit: The paper provides the first complete, self-contained proof that:
$\lim_{n \to \infty} \frac{B^{(8)}_n}{s_n} = \frac{7}{32}\zeta(3)$
Proof of Conjecture Z1: It rigorously proves the Ramanujan Machine conjecture:
$\text{PCF}\left((2n+1)(3n^2+3n+1), -n^6\right) = \frac{8}{7\zeta(3)}$
Explicit Period Polynomial: The derivation of the explicit period polynomial $P_8(\tau) = \frac{7}{32}\zeta(3)(8\tau^2+1)$ for the level-8 case, bridging the gap between the modular form and the specific zeta value.
Methodological Synthesis: The work successfully synthesizes the theory of eta-products, Eichler integrals, and singularity analysis to solve a problem that had resisted complete proof despite numerical evidence.

4. Results

Theorem 1.1: Establishes the limit $\frac{7}{32}\zeta(3)$ for the ratio of the companion sequence to the Apéry-like sequence.
Theorem 1.2: Establishes the convergence of the specific polynomial continued fraction to $\frac{8}{7\zeta(3)}$ .
Explicit Formulas: Provides exact expressions for the modular parametrization $t(\tau)$ , the modular form $g_8(\tau)$ , and the period polynomial $P_8(\tau)$ .

5. Significance

Resolution of Open Problems: This paper resolves a specific conjecture from the "Ramanujan Machine" project, which uses algorithms to discover new mathematical identities. It validates the conjecture with rigorous analytic number theory.
Advancement of Apéry-Limit Theory: While the level-6 case (Domb numbers) was previously solved, the level-8 case is more complex due to the structure of the modular curve and the specific nature of the singularities. This proof extends the methodology to higher levels.
Connection to Zeta Values: The result reinforces the deep connection between combinatorial sequences (defined by recurrences), modular forms, and special values of the Riemann zeta function ( $\zeta(3)$ ).
Computational Verification: The paper moves beyond numerical verification (which had suggested the value) to a theoretical proof, demonstrating the power of combining modular parametrization with asymptotic analysis.

In summary, Alex Shvets provides a definitive proof for a significant conjecture in the field of experimental mathematics, utilizing advanced techniques in modular forms to bridge the gap between discrete recurrence relations and continuous analytic limits.