Closed-Form Evaluation of Arctanh Power Sums via Infinite Products

TL;DR

This paper derives closed-form expressions for infinite arctanh power sums, linking them to gamma functions and zeta values, and explores their properties and applications in number theory.

Contribution

It provides the first closed-form formulas for these sums, connects them to gamma and zeta functions, and analyzes their mathematical properties and applications.

Findings

Derived closed-form expressions for arctanh power sums.

Connected sums to gamma functions and zeta values.

Proved properties like monotonicity and convexity of the sums.

Abstract

We establish closed-form expressions for the infinite series sum from n=2 to infinity of arctanh(n^-k) for all integers k >= 2 by connecting these sums to infinite product formulas involving the gamma function. Our approach uses logarithmic manipulations, the Fubini-Tonelli theorem, and Frullani's integral theorem. As applications, we derive a structural identity relating the Riemann zeta function zeta(k) to these sums, establish a new series representation for the Euler-Mascheroni constant gamma, and show that this representation admits an exponentially convergent reformulation via zeta values. We further prove that h(k) = sum from n=2 to infinity of arctanh(n^-k) is strictly decreasing and strictly convex in k, and we establish explicit two-sided bounds and asymptotic expansions. The decimal expansions of the closed-form values and several auxiliary sequences arising from these identities are cataloged in the OEIS.