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

Luc Ramsès Talla Waffo

TL;DR

The paper studies the Mellin transforms $\Phi_1(s)$ and $\Phi_2(s)$ of the functions $1/\operatorname{arctanh} x$ and $1/(\sqrt{1-x^2}\,\operatorname{arctanh} x)$ on $(0,1)$, highlighting their strong connections to the arithmetic nature of the ratios $\zeta(2n+1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$. It develops contour-integral techniques to evaluate even-integer Mellin transforms, producing closed forms in terms of derivatives $\zeta'(\cdot)$ and $\beta'(\cdot)$, and derives extensive series relations between $\Phi_1$ and $\Phi_2$ that link these transforms to sums of $\zeta$ and $\beta$ values. The work also introduces vanishing combinatorial identities, Malmsten-type integrals related to polylogarithms, and a novel power-series representation for $\Phi_1$ and $\Phi_2$ via coefficients from $1/\operatorname{arctanh}x$ and $1/(\sqrt{1-x^2}\,\operatorname{arctanh}x)$. Collectively, these results provide new analytic tools for hyperbolic integrals and offer fresh perspectives on irrationality questions for the classical constants involved. The combination of residue calculus, asymptotic bounds, and series-transform relations yields a coherent framework to study the arithmetic content of these Mellin transforms and their connections to Dirichlet $\eta$ and $\beta$ functions.

Abstract

We investigate the Mellin transforms of \(1/\operatorname{arctanh} x\) and \(1/(\sqrt{1-x^{2}}\,\operatorname{arctanh} x)\), viewed as compactly supported functions on \((0,1)\). These transforms are closely connected with conjectures on the arithmetic nature of the ratios \(ζ(2n+1)/π^{2n+1}\) and \(β(2n)/π^{2n}\). While their values at odd integers were previously studied, the evaluation at even integers leads to classes of improper integrals that cannot be handled by parity arguments. Using contour integration techniques, we derive explicit closed-form expressions involving derivatives of the Riemann zeta and Dirichlet beta functions, thereby extending earlier results and providing new analytic tools for the study of related hyperbolic integrals.

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

TL;DR

The paper studies the Mellin transforms and of the functions and on , highlighting their strong connections to the arithmetic nature of the ratios and . It develops contour-integral techniques to evaluate even-integer Mellin transforms, producing closed forms in terms of derivatives and , and derives extensive series relations between and that link these transforms to sums of and values. The work also introduces vanishing combinatorial identities, Malmsten-type integrals related to polylogarithms, and a novel power-series representation for and via coefficients from and . Collectively, these results provide new analytic tools for hyperbolic integrals and offer fresh perspectives on irrationality questions for the classical constants involved. The combination of residue calculus, asymptotic bounds, and series-transform relations yields a coherent framework to study the arithmetic content of these Mellin transforms and their connections to Dirichlet and functions.

Abstract

We investigate the Mellin transforms of and \(1/(\sqrt{1-x^{2}}\,\operatorname{arctanh} x)\), viewed as compactly supported functions on \((0,1)\). These transforms are closely connected with conjectures on the arithmetic nature of the ratios \(ζ(2n+1)/π^{2n+1}\) and \(β(2n)/π^{2n}\). While their values at odd integers were previously studied, the evaluation at even integers leads to classes of improper integrals that cannot be handled by parity arguments. Using contour integration techniques, we derive explicit closed-form expressions involving derivatives of the Riemann zeta and Dirichlet beta functions, thereby extending earlier results and providing new analytic tools for the study of related hyperbolic integrals.
Paper Structure (9 sections, 9 theorems, 240 equations)

This paper contains 9 sections, 9 theorems, 240 equations.

Table of Contents

  1. The values at odd integers and some related results
  2. On the integral $\int_0^{+\infty} \dfrac{\sinh^{2q+1}z\,\ln z}{\cosh^n z}\,dz$
  3. On the integral $\int_0^{+\infty} \dfrac{\sinh^{2q}z}{z \, \cosh^{n}z}\,dz$
  4. Vanishing functions with combinatorial coefficients
  5. Some Malmsten's integrals related to the polylogarithm
  6. The Mellin's transforms of $\dfrac{1}{\operatorname{arctanh} x}$ and $\dfrac{1}{\sqrt{1-x^2}\,\operatorname{arctanh} x}$
  7. Bounds and asymptotics
  8. Sums of beta as series of zeta and sums of zeta as series of beta
  9. A new perspective on the arithmetic nature of the ratios $\dfrac{\zeta(2n+1)}{\pi^{2n+1}}$ and $\dfrac{\beta(2n)}{\pi^{2n}}$

Key Result

Lemma 1.1

[lemma]lemma:polynomials Consider the polynomials

Theorems & Definitions (18)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 8 more