The Mellin's transforms of $\frac{1}{\mathrm{arctanh}x}$ and $\frac{1}{\sqrt{1-x^2}\mathrm{arctanh}x}$

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.