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

TL;DR

This paper derives explicit formulas for the Mellin transforms of specific functions involving arctanh, connecting them to deep conjectures about special values of zeta and beta functions, and introduces new analytic methods for hyperbolic integrals.

Contribution

It provides explicit closed-form expressions for Mellin transforms at even integers, extending previous work and introducing contour integration techniques for related hyperbolic integrals.

Findings

Explicit formulas involve derivatives of zeta and beta functions.

New contour integration methods for hyperbolic integrals.

Connections to conjectures on ratios of zeta and beta values.

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 \(\zeta(2n+1)/\pi^{2n+1}\) and \(\beta(2n)/\pi^{2n}\). 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.