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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.06682 |
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| _version_ | 1866914376212021248 |
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| author | Goulden, Ryan |
| author_facet | Goulden, Ryan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06682 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arctanh Sums: Analytic Continuation and Prime-Restricted Theory Goulden, Ryan General Mathematics 11MO6, 11M26 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. |
| title | Arctanh Sums: Analytic Continuation and Prime-Restricted Theory |
| topic | General Mathematics 11MO6, 11M26 |
| url | https://arxiv.org/abs/2603.06682 |