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Perspectives on the arithmetic nature of the ratios $ζ(2n + 1)/π^{2n+1}$ and $β(2n)/π^{2n}$

This paper unifies various analytic frameworks to study the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, thereby motivating linear independence conjectures that would establish the irrationality of the ratios $\zeta(2n + 1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$ .

Original authors: Luc Ramsès Talla Waffo

Published 2026-01-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luc Ramsès Talla Waffo

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 a detective trying to solve a mystery about numbers. For centuries, mathematicians have been fascinated by two special families of numbers: the Riemann zeta function (let's call it Zeta) and the Dirichlet beta function (let's call it Beta).

Here is the puzzle:

When you plug in even numbers (like 2, 4, 6) into Zeta, the answers are beautiful and predictable. They are always a simple fraction multiplied by powers of $\pi$ (the ratio of a circle's circumference to its diameter). It's like finding a perfect recipe every time.
When you plug in odd numbers (like 3, 5, 7) into Zeta, or even numbers into Beta, the answers become a complete mystery. We don't have a simple recipe. We suspect these answers are "irrational" (meaning they can't be written as a simple fraction) and that they don't fit neatly into the pattern of $\pi$ . But nobody has proven it yet.

This paper, written by Luc Ramsès Talla Waffo, is like a new map for the detective. Instead of trying to guess the numbers directly, the author builds a series of mathematical bridges (called integrals) that connect these mysterious numbers to a specific, strange-looking shape involving logarithms and trigonometry.

The "Magic Bridge" Analogy

Think of the mysterious numbers (like $\zeta(3)$ or $\beta(2)$ ) as locked treasure chests. We know they are inside, but we can't open them to see what's inside directly.

The author's main idea is to build a bridge made of a specific mathematical formula (an integral) that leads from the treasure chest to a known location.

The author shows that if you calculate a specific area under a curve (involving things like $\sin(4x)$ and $\ln(\tan x)$ ), the result is a mixture of these mysterious numbers.
It's like saying: "If you mix 7 parts of Mystery Number A with 3 parts of Mystery Number B, you get this specific amount of 'Bridge Material'."

The "Recipe" Discovery

The paper does a lot of heavy lifting to show that these "Bridge Materials" aren't random. They follow a strict pattern:

The Bridge Formula: The author calculates several of these bridges for different numbers.
The Pattern: They realize that every bridge is just a combination of the mysterious numbers ( $\zeta(3), \zeta(5), \zeta(7)$ , etc.) multiplied by powers of $\pi$ .
The Reverse Engineering: Because we know how to build the bridges, we can work backward. The author proves that you can express the mysterious numbers as a combination of these bridges.

The Big Guess (The Conjecture)

Now comes the exciting part. The author proposes a bold guess (a conjecture) to solve the mystery of whether these numbers are "irrational."

The Analogy:
Imagine you have a bag of ingredients (the mysterious numbers). You want to know if you can make a specific dish (a rational number) using only these ingredients and a pinch of salt ( $\pi$ ).

The author suggests: "If you take all the 'Bridge Materials' we built, mix them together in any way you want using simple fractions, you will never accidentally create the number $\pi$ itself."

If this guess is true, it proves that the mysterious numbers (like $\zeta(3)/\pi^2$ ) are not simple fractions. They are "wild" numbers that cannot be tamed by simple arithmetic.

The "Kitchen" of Integrals

The paper lists a whole kitchen full of different recipes (integrals) that all lead to the same result. Some look like:

Integrating a wavy line ( $\sin$ ) against a logarithmic curve.
Integrating a hyperbolic function ( $\sinh$ ) over a hyperbolic cosine ( $\cosh$ ).
Integrating complex shapes involving "polylogarithms" (a fancy type of function).

The author shows that all these different recipes are actually just different ways of saying the same thing. They are all connected to the same underlying structure.

The "Fourier" Connection

Towards the end, the author tries to prove their guess is likely correct by looking at a famous musical concept called Fourier series (which breaks complex sounds into simple notes).

They take a known sound wave ( $\ln(\tan x)$ ) and break it down.
They try to reconstruct the number $\pi$ using the "Bridge Materials."
Their calculations suggest that you cannot reconstruct $\pi$ from these materials. This lack of "reconstruction" is a strong hint that the author's guess is right: the mysterious numbers are indeed independent and irrational.

Summary

In simple terms, this paper doesn't solve the mystery of these numbers (it doesn't prove they are irrational yet). Instead, it:

Builds new tools (integrals) to measure these numbers.
Shows the tools are connected in a very specific, linear way.
Proposes a test: If these tools can't be mixed to create $\pi$ , then the numbers we are studying are definitely irrational.

It's a work of structural engineering for numbers, laying the foundation for future mathematicians to finally prove the long-standing mystery of these "odd" and "even" function values.

Technical Summary: Perspectives on the arithmetic nature of the ratios $\zeta(2n + 1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$

Problem Statement
The paper addresses the long-standing open problem regarding the arithmetic nature of the Riemann zeta function at odd integers, $\zeta(2n+1)$ , and the Dirichlet beta function at even integers, $\beta(2n)$ . While Euler established explicit formulas for $\zeta(2n)$ and $\beta(2n+1)$ involving rational multiples of powers of $\pi$ , no such closed-form expressions exist for $\zeta(2n+1)$ or $\beta(2n)$ . The central question is whether the ratios $\zeta(2n+1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$ are rational. The author notes that despite numerous attempts, no rational coefficients have been found, and heuristic evidence suggests these ratios are irrational, though a rigorous proof remains elusive.

Methodology
The investigation relies on deriving and analyzing specific integral representations for these values. The methodology proceeds through the following steps:

Integral Derivation: The author utilizes Malmsten-type integrals involving logarithmic kernels ( $\ln \ln(1/x)$ ) and polylogarithms ( $\text{Li}_s$ ) to express the target ratios. Key starting points include integrals evaluated by Blagouchine and Adamchik, which are generalized to higher orders.
Transformation and Reduction: Through variable substitutions (e.g., $x \to \tan u$ ) and integration by parts, the author transforms integrals over $(0, 1)$ into integrals over $(0, \pi/4)$ involving trigonometric functions and $\ln(\tan x)$ .
Linear Algebraic Framework: The paper employs Gauss elimination on systems of these integral equations. By establishing linear relationships between integrals of the form $\int_0^{\pi/4} \frac{\sin(4nx)}{\ln(\tan x)} dx$ and $\int_0^{\pi/4} \frac{\cos((4n-2)x)}{\ln(\tan x)} dx$ , the author demonstrates that these integrals can be expressed as finite linear combinations of the ratios $\zeta(2p+1)/\pi^{2p}$ and $\beta(2p)/\pi^{2p-1}$ with rational coefficients.
Contour Integration: Theoretical justification for the structure of these integrals is provided via contour integration techniques on the complex plane, specifically analyzing residues of functions involving $\text{sech}^n z$ and $\sinh((2k+1)z)/z$ .
Formal Proofs: Propositions 2.1 and 2.2 provide formal derivations linking specific polylogarithmic integrals to the zeta and beta ratios, utilizing properties of the Dirichlet $\eta$ function and functional equations of the $\beta$ function.

Key Contributions and Results

Unified Integral Representations: The paper presents a comprehensive table of integrals that evaluate to linear combinations of $\zeta(2p+1)/\pi^{2p}$ $ζ (2 p + 1) / π^{2 p}$ and $\beta(2p)/\pi^{2p-1}$ $β (2 p) / π^{2 p - 1}$ . These include:
- Integrals over $(0, \infty)$ involving hyperbolic functions: $\int_0^\infty \frac{\sinh((2k+1)x)}{x \cosh^n x} dx$ .
- Integrals over $(0, 1)$ involving polylogarithms and double logarithms: $\int_0^1 \frac{\text{Li}_{-2n-1}(-x^2)}{x} \ln \ln(1/x) dx$ and $\int_0^1 \frac{\text{Im}[\text{Li}_{-2n}(ix)]}{x} \ln \ln(1/x) dx$ .
- Trigonometric integrals over $(0, \pi/4)$ : $\int_0^{\pi/4} \frac{\sin(4nx)}{\ln(\tan x)} dx$ .
Explicit Coefficients: The author derives explicit rational coefficients for these linear combinations. For instance, the paper shows that $\int_0^{\pi/4} \frac{\sin(4x)}{\ln(\tan x)} dx = -7\zeta(3)/\pi^2$ and provides higher-order expansions for $\sin(8x), \sin(12x)$ , etc., involving $\zeta(3), \zeta(5), \zeta(7)$ , etc.
Polynomial Formulations: The paper demonstrates that $\zeta(2n+1)/\pi^{2n}$ and $\beta(2n)/\pi^{2n-1}$ can be represented as integrals of the form $\int_0^1 \frac{x P_n(x)}{\text{arctanh}(x)} dx$ and $\int_0^1 \frac{x P_n(x)}{\sqrt{1-x^2}\text{arctanh}(x)} dx$ , respectively, where $P_n(x)$ are specific polynomials with rational coefficients.
Conjectures on Linear Independence: The primary result is the formulation of conjectures linking the irrationality of the target ratios to the linear independence of $\pi$ over the field of rational numbers ( $\mathbb{Q}$ ) with respect to the set of these integral values. Specifically, the paper posits that if $\pi$ is not in the $\mathbb{Q}$ -linear span of the sequence of these integrals, then the ratios $\zeta(2n+1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$ are irrational.

Significance and Claims
The paper claims to provide a "unifying perspective" on these arithmetic problems by collecting various analytic frameworks that lead to the same values. Its significance lies in:

Reformulation of the Problem: It shifts the focus from finding explicit rational coefficients (which has proven futile) to proving the linear independence of $\pi$ from a specific set of integrals.
Structural Insight: By identifying that diverse integrals (hyperbolic, trigonometric, polylogarithmic) all reduce to the same linear combinations of zeta and beta ratios, the paper suggests a deep underlying structural pattern.
Pathway to Irrationality: The author argues that proving the conjecture $\pi \notin \text{Span}_{\mathbb{Q}}\{I_1, I_2, \dots\}$ would immediately imply the irrationality of $\zeta(2n+1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$ .

Limitations and Modesty
The author explicitly acknowledges the limitations of the current work:

Rigor: The proofs in Propositions 2.1 and 2.2 are noted to lack full mathematical rigor in specific steps, particularly regarding the interchange of summation and integration and the handling of regularized versions of divergent series. The author states that a rigorous derivation of the coefficients requires further detailed work found in reference [10].
Heuristic Nature: The paper admits that the impossibility of finding rational coefficients is currently supported by "heuristic methods and deep computations by brute force" rather than a strong proof.
Conjectural Status: The central claims regarding irrationality remain conditional on the truth of the proposed linear independence conjectures. The paper does not claim to have proven the irrationality of these ratios but rather offers a new framework for attempting to do so.
Fourier Series Identity: The final section presents a formal identity derived from Fourier series that supports the conjecture, but the author explicitly notes the "weakness and lack of rigor" in the convergence arguments used to derive it.

In summary, the paper serves as a collection of analytic tools and a proposal for a new strategy to tackle the irrationality of odd zeta values and even beta values, framing the problem in terms of the linear independence of $\pi$ from a specific family of integrals.