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The ratio of Kummer's first factor of the class number of the cyclotomic number field $\\mathbb{Q}(\\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\\log q)$ products and $O(q)$ logarithms. Using it we obt"},"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":"1908.01152","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-03T11:44:41Z","cross_cats_sorted":[],"title_canon_sha256":"ca98e785a6cad3c2bbb4c899f9a43ee295098cb14f35cc95f12285cc2693dea6","abstract_canon_sha256":"c946027c1592a42710511e24138d0351bfa0e65120249f1618ac43979a837b12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:13:16.016666Z","signature_b64":"42aahgQU2pqxEojEU2xcupg21em518bd4FrSKL/jdHIDcVf3FRG8xyTifrNvMoeeiyYABWkj/APNi1RL/ilBCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62f244800014d56490b5bb87b6538baabb6983b2b23f6d9fde3364a710c01f87","last_reissued_at":"2026-07-05T00:13:16.016322Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:13:16.016322Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computation of the Kummer ratio of the class number for prime cyclotomic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alessandro Languasco, Alisa Sedunova, Pieter Moree, Sumaia Saad Eddin","submitted_at":"2019-08-03T11:44:41Z","abstract_excerpt":"Let $\\zeta_q$ be a primitive $q^{\\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\\mathbb{Q}(\\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\\log q)$ products and $O(q)$ logarithms. 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