Arctanh Sums: Analytic Continuation and Prime-Restricted Theory

TL;DR

This paper explores the analytic properties of arctanh sums related to prime numbers, extending their domain, analyzing their poles and zeros, and establishing connections with the Riemann zeta function and prime number theory.

Contribution

It develops the analytic continuation of arctanh sums, derives their Laurent expansions, and introduces prime-restricted analogues with implications for transcendence and zero distributions.

Findings

h(k) extends meromorphically with simple poles at specific points

h has exactly one real zero in each inter-polar interval

Prime-restricted h_p(k) shows a pi-cancellation mechanism and a zero product formula

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.