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A Modular Form Proof of the Irrationality of $\zeta\left(3\right)$

This paper presents an expository proof of the irrationality of $\zeta(3)$ by constructing a specific Eichler integral from modular forms of level 6 to generate a power series that satisfies Beukers' irrationality criterion.

Original authors: Pang Ern Thang

Published 2026-07-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pang Ern Thang

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 Mystery of the Unbreakable Number

Imagine you are a detective trying to solve a puzzle about numbers. In the world of mathematics, there is a special family of numbers called the "Riemann zeta function." Think of this function as a giant, magical calculator that takes a number, say $s$ , and adds up an infinite list of fractions: $1 + 1/2^s + 1/3^s + 1/4^s$ , and so on, forever. When you plug in even numbers like 2 or 4, the calculator gives you answers that are well-behaved and can be written as simple fractions involving $\pi$ (the ratio of a circle's circumference to its diameter). But when you plug in odd numbers like 3, 5, or 7, the calculator gets weird. The answers don't seem to follow a simple pattern.

The big question is: Are these odd answers "rational" (meaning they can be written as a simple fraction like $3/4$ ) or "irrational" (meaning they are messy, never-ending decimals that can never be written as a fraction, like $\pi$ or $\sqrt{2}$ )? For a long time, mathematicians knew the even answers were irrational, but the odd ones were a mystery. One of the most famous mysteries is the answer for $s=3$ , a number known as Apéry's constant. If this number is irrational, it means it is a unique, unbreakable piece of the mathematical universe that cannot be simplified. Proving this is like showing that a specific lock has no key that fits perfectly.

The Paper's Journey: A Modular Map to the Truth

This paper, written by Pang Ern Thang, presents a fresh and elegant way to prove that Apéry's constant, $\zeta(3)$ , is indeed irrational. While the original proof by Roger Apéry in 1978 was a brilliant but somewhat mysterious feat, this author shows that the proof is actually a natural consequence of a different branch of mathematics called "modular forms."

To understand the paper's method, imagine the complex world of numbers as a vast, foggy landscape. Usually, if you try to walk through this landscape, you hit a wall (a "branching value") that stops you from going further. In math terms, this wall limits how far you can extend a function before it breaks down or becomes unpredictable. The author's strategy is to build a special "bridge" using modular forms—functions that have a unique, symmetrical beauty, like a kaleidoscope that looks the same no matter how you rotate it.

The paper constructs a specific bridge using a modular form of "level 6." Think of this level as a specific set of rules for how the kaleidoscope rotates. By using these rules, the author creates a special path (an "Eichler integral") that connects the messy world of $\zeta(3)$ to the symmetrical world of modular forms. The magic happens when they look at the "radius of convergence" of this path. In simple terms, this is how far you can walk along the path before hitting a wall.

Usually, the first wall you hit is quite close. However, because of the special symmetry of the modular forms used in this paper, the path doesn't just stop at the first wall. It magically extends past it, reaching a much larger distance before hitting the next wall. This "extra room" is the key. The paper uses a criterion (a test for irrationality) developed by Beukers, which says: if you can build a path that goes far enough and has a specific pattern of numbers, then the number you are studying must be irrational.

The author calculates that the path extends to a distance of $17 + 12\sqrt{2}$ , which is roughly 33.97. This distance is large enough to pass the test. The paper shows that the numbers along this path have denominators (the bottom numbers of fractions) that are controlled and predictable. Because the path is so long and the numbers are so well-behaved, the test confirms that $\zeta(3)$ cannot be a simple fraction.

In essence, the paper takes a difficult problem about a single number and solves it by showing that the number is part of a larger, symmetrical structure. It's like proving a specific brick is unique not by examining the brick itself, but by showing it is part of a magnificent, unbreakable cathedral. The paper doesn't just guess; it provides a rigorous, step-by-step proof that $\zeta(3)$ is irrational, using the hidden symmetries of the mathematical universe to do the heavy lifting.

Technical Summary: A Modular Form Proof of the Irrationality of $\zeta(3)$

Problem Statement
The paper addresses the irrationality of the Riemann zeta function at odd positive integers, specifically $\zeta(3)$ (Apéry's constant). While Euler established the irrationality of $\zeta(2n)$ for all $n \in \mathbb{N}$ , the irrationality of $\zeta(2n+1)$ remains largely open, with only $\zeta(3)$ proven irrational by Apéry in 1978. The paper seeks to re-derive Apéry's result using the framework of complex analysis on spaces of modular forms, specifically those of level 6, rather than relying on Apéry's original combinatorial recurrence relations.

Methodology
The proof strategy relies on Beukers' irrationality criterion, which requires constructing a specific power series with controlled denominators and a sufficiently large radius of convergence. The methodology proceeds through the following steps:

Modular Forms and Dirichlet Series: The authors utilize modular forms of level 6, specifically the congruence subgroup $\Gamma_1(6)$ . They define a modular function $t(\tau)$ derived from eta quotients and a modular form $F(\tau)$ of weight 4 in the space of cusp forms $S_4(\Gamma_1(6))$ .
Eichler Integrals: An Eichler integral $f(\tau)$ is constructed from $F(\tau)$ , defined as a series $\sum a_n n^{-3} q^n$ . This integral is related to the Dirichlet series $L(F, s)$ , where it is shown that $L(F, 3) = \zeta(3)$ .
Functional Equations: Using the Fricke involution $\tau \mapsto -1/(6\tau)$ , the authors establish a functional equation for the Eichler integral. They construct an auxiliary function $H(\tau) = E(\tau)(f(\tau) - \zeta(3))$ , where $E(\tau)$ is a specific modular form of weight 2. Crucially, $H(\tau)$ is shown to be invariant under the Fricke involution.
Change of Variable: The modular function $t(\tau)$ serves as a coordinate change, mapping a fundamental domain of the extended group $\Gamma_1(6)^+$ univalently onto the upper half-plane. The function $H$ is re-expressed as a power series in $t$ , denoted $H(t)$ .
Radius of Convergence Analysis: The radius of convergence of $H(t)$ is determined by the branching values of the inverse map $\tau(t)$ . The first branching value corresponds to the fixed point of the involution, $\tau = i/\sqrt{6}$ , yielding $t_1 = (\sqrt{2}-1)^4$ . However, because $H(\tau)$ takes the same value on the two local branches of $\tau(t)$ near this point, the apparent singularity is removable. The actual radius of convergence is determined by the next branching value, $t_2 = (\sqrt{2}+1)^4 = 17 + 12\sqrt{2}$ .
Denominator Control: The coefficients of the power series for $H(t)$ are analyzed. The coefficients of $f(t)$ have denominators dividing $D_n^3$ (where $D_n = \text{lcm}(1, \dots, n)$ ), and the coefficients of $E(t)$ are integers. This ensures the series satisfies the rationality and denominator conditions required by Beukers' criterion.

Key Contributions and Results

Construction of $t(\tau)$ : The paper explicitly constructs a modular function $t(\tau)$ for $\Gamma_1(6)$ that maps a specific fundamental domain to the upper half-plane, with known values at cusps and interior points.
Removable Singularity: A critical technical contribution is the demonstration that the function $H(t)$ , constructed from the Eichler integral and the modular form $E$ , extends analytically past the first branching value $t_1$ . This enlargement of the radius of convergence from $|t_1|$ to $|t_2|$ is the mechanism that allows the irrationality proof to succeed.
Proof of Irrationality: By applying Beukers' criterion with the constructed series $H(t)$ , the authors show that the radius of convergence $\rho = 17 + 12\sqrt{2}$ satisfies $\rho > e^3$ (since $d=1, r=3$ ). This inequality forces the conclusion that $\zeta(3)$ is irrational.

Significance and Claims
The paper claims to present an expository proof of the irrationality of $\zeta(3)$ that frames Apéry's result as a "nice consequence of complex analysis on spaces of certain modular forms." The authors emphasize that while this modular form approach successfully recovers the irrationality of $\zeta(3)$ , it does not currently yield results for other odd zeta values (e.g., $\zeta(5)$ ). The work serves to illustrate the deep connection between Apéry's combinatorial proof and the theory of modular forms, specifically highlighting how the geometry of modular curves and the properties of Eichler integrals can be leveraged to establish irrationality. The paper does not propose new applications or future extensions beyond this specific derivation.

The Mystery of the Unbreakable Number

The Paper's Journey: A Modular Map to the Truth

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