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Then $\\langle B \\rangle$ is a free $\\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $\\langle B \\rangle$ with given finite index. Thus, the formal Dirichlet series $\\zeta_{\\langle B \\rangle}(s)=\\sum_{n\\geq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $\\langle B \\rangl"},"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":"1608.08508","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T15:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"dc9e5ea4ee89456fc155e06fe0081a9152d15c0ed84f9bfe71f2a6b47d36d189","abstract_canon_sha256":"602c37089f0cf1a51c65a79182be6556e1a41eadd63f75082d80a9ca4dd80db5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:17.955065Z","signature_b64":"gAKr29jK7fMQMSns7/aaVB9qbD2+KD8YcBIhQm6sn3HnBTbSzjzTNyvNGo/kE1LuVkD85TP4xchHvp1vNvJ9Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37638d6b91b26e0c31d1c910adefb428e772988f8ef6343986c7525fe8d47b1c","last_reissued_at":"2026-05-18T01:07:17.954527Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:17.954527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of ideals of $\\mathbb{Z}[x]$ containing $x(x-\\alpha)(x-\\beta)$ with given index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mitsugu Hirasaka, Semin Oh","submitted_at":"2016-08-30T15:36:43Z","abstract_excerpt":"It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. 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