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The Mellin's transforms of $\dfrac{1}{\operatorname{arctanh} x}$ and $\dfrac{1}{\sqrt{1-x^2}\,\operatorname{arctanh} x}$

This paper employs contour integration to derive explicit closed-form expressions involving derivatives of the Riemann zeta and Dirichlet beta functions for the Mellin transforms of $1/\operatorname{arctanh} x$ and $1/(\sqrt{1-x^2}\,\operatorname{arctanh} x)$ at even integers, thereby extending previous results and offering new analytic tools for investigating the arithmetic nature of specific zeta and beta function ratios.

Original authors: Luc Ramsès Talla Waffo

Published 2026-01-27

📖 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. Specifically, you are investigating two very special, elusive numbers: the Riemann Zeta function (let's call it Z) and the Dirichlet Beta function (let's call it B).

Mathematicians have long suspected that certain ratios involving these numbers and the famous circle constant, Pi ( $\pi$ ), are "irrational." This means they cannot be written as a simple fraction (like 1/2 or 22/7). They are infinite, non-repeating decimals that never settle down.

This paper, written by Luc Ramsès Talla Waffo, is like a new toolkit designed to help crack this case. Here is how the author approaches the problem, explained in everyday terms:

1. The Two Suspects (The Functions)

The author focuses on two specific mathematical "machines" (functions) that take an input number and spit out a result.

Machine A: Takes a number, divides it by the "inverse hyperbolic tangent" (a fancy way of saying it reverses a specific curve), and integrates it.
Machine B: Does the same thing but adds a "curved wall" (the square root of $1-x^2$ ) to the denominator.

Think of these machines as sieves. You pour a stream of numbers into them, and they filter out specific patterns. The author calls these "Mellin transforms," which is just a fancy name for a specific type of mathematical sieve that turns a function into a new one based on its "shape."

2. The Old vs. The New Clue

Previously, mathematicians knew how to use these sieves when the input was an odd number (like 1, 3, 5). It was like knowing how to open a door with a left-handed key.

However, this paper tackles the even numbers (2, 4, 6).

The Problem: When you try to use the old "left-handed" methods on even numbers, the math breaks down. It's like trying to open a right-handed door with a left-handed key; the mechanism just doesn't turn, or worse, it gives you a result of zero, which tells you nothing.
The Solution: The author invents a new set of tools using contour integration. Imagine walking around a mountain (a complex path in the mathematical landscape) instead of trying to climb straight up. By walking around the "poles" (the tricky spots where the math explodes), the author can calculate the value of these even-numbered inputs without them vanishing into nothingness.

3. The Treasure Map (The Results)

By using this new "walking around the mountain" technique, the author derives explicit formulas.

Instead of just getting a messy integral that can't be solved, the author finds a closed-form expression.
Think of this as finding a treasure map that says, "The treasure isn't buried in a random hole; it's exactly 3 feet under the oak tree, mixed with a specific amount of gold dust (derivatives of Zeta and Beta functions)."

The paper provides a long list of these "maps" for various even numbers. For example, it shows exactly how to calculate the integral for $x^2$ , $x^4$ , $x^6$ , etc., in terms of known mathematical constants and their derivatives.

4. The Big Picture: Why Does This Matter?

The ultimate goal isn't just to calculate these numbers for fun. It's about proving they are irrational.

The author suggests a clever strategy:

If you can prove that the results from these "sieves" (the Mellin transforms) cannot be combined to create the number $\pi$ , then you prove that the ratios of Zeta and Beta to $\pi$ are irrational.
The paper shows that these transforms can be written as infinite series involving combinations of Zeta and Beta values.
It essentially says: "If we can show that these specific combinations of numbers never add up to $\pi$ , then the mystery of their irrationality is solved."

5. The "Vanishing" Act

One of the cooler tricks in the paper involves "vanishing functions." The author identifies specific mathematical patterns (combinatorial coefficients) that, when added up, magically cancel each other out to zero.

Analogy: Imagine a group of people holding signs with numbers on them. If you ask them to stand in a specific formation, the total sum of the numbers on the signs becomes zero. The author uses these "zero-sum" groups to simplify complex equations, stripping away the noise to reveal the core signal.

Summary

In short, this paper is a mathematical repair manual.

The Problem: We couldn't calculate the "even" values of two important mathematical sieves because our old tools failed.
The Fix: The author built a new tool (contour integration) to walk around the obstacles and get the answers.
The Result: We now have exact formulas for these values.
The Goal: These formulas provide the necessary evidence to potentially prove that certain famous number ratios are irrational, solving a long-standing puzzle in number theory.

The paper doesn't claim to cure diseases or build bridges; it stays strictly within the realm of pure mathematics, offering new ways to see the hidden structure of numbers like $\pi$ , Zeta, and Beta.

Technical Summary: The Mellin Transforms of $1/\text{arctanh } x$ and $1/(\sqrt{1-x^2}\text{arctanh } x)$

Problem Statement
This work investigates the Mellin transforms of two specific functions, viewed as compactly supported on the interval $(0, 1)$ :
$\Phi_1(s) = \int_0^1 \frac{x^{s-1}}{\text{arctanh } x} \, dx \quad \text{and} \quad \Phi_2(s) = \int_0^1 \frac{x^{s-1}}{\sqrt{1-x^2}\text{arctanh } x} \, dx$
These transforms are intrinsically linked to the arithmetic nature of the ratios $\frac{\zeta(2n+1)}{\pi^{2n+1}}$ and $\frac{\beta(2n)}{\pi^{2n}}$ . Previous research (cited as [2], [3]) established integral representations for these ratios involving even polynomials. However, a gap remained regarding the evaluation of these transforms at odd integers (corresponding to even powers of $x$ in the integrand). While values at even integers were previously studied, the odd integer cases lead to improper integrals that cannot be resolved using standard parity arguments (as the integrands are odd functions), necessitating a new analytical approach.

Methodology
The paper employs a combination of combinatorial analysis, series expansion, and complex analysis techniques, specifically contour integration.

Combinatorial and Series Approaches (Sections 1 & 6):
- The author defines specific polynomials $P_n(x)$ and $Q_n(x)$ related to the roots of hyperbolic functions to derive coefficients for Taylor expansions.
- By substituting $x = \tanh u$ , the original integrals are transformed into integrals over $(0, \infty)$ involving $\tanh$ , $\cosh$ , and $\ln u$ .
- Binomial expansions and geometric series are utilized to express these integrals as sums involving derivatives of the Dirichlet eta function ( $\eta'$ ) and the Dirichlet beta function ( $\beta'$ ).
- Section 6 establishes coupled systems relating $\Phi_1(s)$ and $\Phi_2(s)$ via binomial expansions of $\sqrt{1-x^2}$ and $1/\sqrt{1-x^2}$ , allowing the expression of sums of beta values as series of zeta values and vice versa.
Contour Integration (Sections 2 & 3):
- To rigorously evaluate the integrals at odd integers (which correspond to integrals of the form $\int_0^\infty \frac{\sinh^{2q+1} z \ln z}{\cosh^n z} dz$ ), the author utilizes a rectangular contour in the complex plane.
- Unlike previous works that handled even functions by extending the domain to $(-\infty, \infty)$ , this work addresses the odd nature of the integrand. The method involves integrating $z^2 f_{q,n}(z)$ around a rectangle $[-R, R] \times [0, i\pi]$ and applying Cauchy's Residue Theorem.
- The poles of the integrand are identified, and residues are calculated using Cauchy's integral formula and derivatives of $\ln^2 z$ .
- This technique yields closed-form expressions involving derivatives of the Riemann zeta function ( $\zeta'$ ) and the Dirichlet beta function ( $\beta'$ ), along with logarithmic terms ( $\ln \pi, \ln 2$ ).

Key Contributions and Results

Explicit Closed-Form Expressions: The paper derives explicit formulas for $\Phi_1(2n+1)$ $Φ_{1} (2 n + 1)$ and $\Phi_2(2n+1)$ $Φ_{2} (2 n + 1)$ . These are expressed as finite linear combinations of derivatives of the eta and beta functions at negative integers (e.g., $\eta'(-2i-1)$ $η^{'} (- 2 i - 1)$ and $\beta'(-2i)$ $β^{'} (- 2 i)$ ).
- Example: $\int_0^1 \frac{x^2}{\text{arctanh } x} dx = \frac{4}{3}\eta'(-1) + \frac{8}{3}\eta'(-3)$ .
Evaluation of Improper Integrals: The work provides a systematic method for evaluating integrals of the form $\int_0^\infty \frac{\sinh^{2q+1} z \ln z}{\cosh^n z} dz$ and $\int_0^\infty \frac{\sinh^{2q} z}{z \cosh^n z} dz$ . These results are presented in terms of $\zeta'$ and $\beta'$ values, Bernoulli numbers, and Euler numbers.
Vanishing Combinatorial Identities: Section 4 identifies specific classes of functions with combinatorial coefficients that vanish for certain integer arguments, which are crucial for simplifying the residue calculations.
Asymptotic Analysis: The paper establishes bounds and asymptotic expansions for $\Phi_1(s)$ and $\Phi_2(s)$ as $s \to 1^+$ , providing constants $C_1$ and $C_2$ expressed in terms of $\zeta'(2)$ , $\beta'(1)$ , and logarithmic terms.
Reformulation of Irrationality Conjectures: In Section 6.3, the author reformulates the conjectures regarding the irrationality of $\frac{\zeta(2n+1)}{\pi^{2n+1}}$ and $\frac{\beta(2n)}{\pi^{2n}}$ . The irrationality of these ratios is shown to follow if specific infinite series involving the coefficients of the Taylor expansions of $1/\text{arctanh } x$ and $1/(\sqrt{1-x^2}\text{arctanh } x)$ do not span the rational numbers in a way that allows $\pi$ to be represented.

Significance and Claims
The paper claims three primary contributions:

Arithmetic Insight: It studies Mellin transforms deeply connected to the arithmetic nature of $\zeta(2n+1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$ , extending previous results to odd integer arguments which were previously unaddressed due to parity constraints.
New Perspectives on Integrals: It provides new perspectives on integrals that may have been overlooked in the works of Blagouchine, Adamchik, and others, specifically those involving odd powers of hyperbolic sine combined with logarithmic terms.
Analytic Tools: It develops complex analysis techniques (specifically the contour integration method for odd integrands) that yield non-trivial identities useful in combinatorics and the evaluation of hyperbolic integrals.

The author notes that while the results at even integers were known, the derivation at odd integers required a distinct method because parity arguments fail (yielding trivial zero if applied naively). The derived closed-form expressions involving derivatives of L-functions provide new analytic tools for studying related hyperbolic integrals and the arithmetic properties of the aforementioned ratios.