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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1806.07762","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-20T14:29:43Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"a65b74d3292aec839fbdfc9e57f7505ba416835d6bbc5caa215b87185a1b4480","abstract_canon_sha256":"07844d40df9bd6a8615eac4cdbedb1be46e7302c6b4c85b2e0322a0a1f25b50f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:08.742110Z","signature_b64":"A7/8S+SyZO6jY+MQJtWyKOe/Fwb2PjoCjWCMxP+KWf7bx49OMFqBc2g4rCYTCBdR8/zEsKgQhtvYrPV9HXTqBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b10af80308ef19c4bf1393a80d024e0bef1e0d96934ef1298f6cf3ae243e00a4","last_reissued_at":"2026-05-17T23:42:08.741394Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:08.741394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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