Arctanh Sums: Analytic Continuation and Prime-Restricted Theory

Abstract

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.