{"paper":{"title":"Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for $\\zeta(2m+1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Atul Dixit, Bibekananda Maji, Rahul Kumar, Rajat Gupta","submitted_at":"2018-01-28T05:25:57Z","abstract_excerpt":"A comprehensive study of the generalized Lambert series $\\displaystyle\\sum_{n=1}^{\\infty}\\frac{n^{N-2h}\\exp{(-an^{N}x)}}{1-\\exp{(-n^{N}x)}}, 0<a\\leq 1,\\ x>0$, $N\\in\\mathbb{N}$ and $h\\in\\mathbb{Z}$, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan's famous formula for $\\zeta(2m+1)$, $m>0$ and the transformation formula for $\\log\\eta(z)$. Numerous important special cases of our transformations are derived. An identity relating $\\zeta(2N+1), \\zeta(4N+1),\\cdots, \\zeta(2Nm+1)$ is obtained for $N$ odd and $m\\in\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09181","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}