arXiv subjects

Ryan Goulden

Publications and source records attributed to Ryan Goulden.

2 recordsLinked to original sources

arXiv2026

Arctanh Sums: Analytic Continuation and Prime-Restricted Theory

We study the arctanh sums h(k) = sum_{n=2}^\infty arctanh(n^{-k}) as a function of a complex variable k. Building on the closed-form identity h(k) = (1/2) log(g(2k)/g(k)^2) (proved in the companion preprint arXiv:2602.06244), we develop the analytic continuation and prime-restricted multiplicative theory. We prove that h extends meromorphically to Re(k) > 0 with simple poles at k = 1/(2m+1), derive Laurent expansions at its poles (including k = 1), and obtain a Mittag-Leffler decomposition encoding the Dirichlet lambda function. We also show that h has exactly one simple real zero in each inter-polar interval. Finally, for the prime-restricted analogue h_p(k) = log(zeta(k)) - (1/2) log(zeta(2k)), we establish a pi-cancellation mechanism implying unconditional transcendence of h_p(2j), and derive a product formula over the nontrivial zeros of zeta with O(|Im(rho)|^{-2}) decay.

math.GM↗

arXiv2026

Closed-Form Evaluation of Arctanh Power Sums via Infinite Products

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.

math.GM↗