{"paper":{"title":"On Dirichlet's lambda functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Min-Soo Kim, Su Hu","submitted_at":"2018-06-20T14:29:43Z","abstract_excerpt":"Let $$\\lambda(s)=\\sum_{n=0}^\\infty\\frac1{(2n+1)^s},$$ $$\\beta(s)=\\sum_{n=0}^\\infty\\frac{(-1)^{n}}{(2n+1)^s},$$ and $$\\eta(s)=\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n^s}$$ be the Dirichlet lambda function, its alternating form, and the Dirichlet eta function, respectively. According to a recent historical book by Varadarajan (\\cite[p.~70]{Varadarajan}), these three functions were investigated by Euler under the notations $N(s)$, $L(s)$, and $M(s)$, respectively.\n  In this paper, we shall present some additional properties for them. That is, we obtain a number of infinite families of linear recurren"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07762","kind":"arxiv","version":4},"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"}