{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:4HGJDZDOTZ725JYSCI7Y2PQKW7","short_pith_number":"pith:4HGJDZDO","schema_version":"1.0","canonical_sha256":"e1cc91e46e9e7faea712123f8d3e0ab7df9dd55ad79b181ae036a0643d726697","source":{"kind":"arxiv","id":"2508.08996","version":1},"attestation_state":"computed","paper":{"title":"Unconditional results for Artin-type problems over number fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pietro Sgobba","submitted_at":"2025-08-12T15:03:43Z","abstract_excerpt":"Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\\times$. For all but finitely many primes $\\mathfrak p$ of $K$, the reduction $(G \\bmod \\mathfrak p)$ generates a well-defined subgroup of the multiplicative group of the residue field at $\\mathfrak p$, and we may consider its index. We study the primes of $K$ for which this index lies in a given set of positive integers $S$. In particular, we prove that under certain convergence conditions on series associated to $S$ this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provi"},"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":"2508.08996","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-08-12T15:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"e034fca417b1faf7e234dd4a7299d06acc21bb20367e315e9cf5795f80811583","abstract_canon_sha256":"f3cc8dfbeeff0e057c4eef11bcaf16c8e026a2ee3e6fe4d66d2a2f7a79160d97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:52:38.142640Z","signature_b64":"qjiJwGJgsZp5HIvh/1YiMFEjWtC8CmneTixdsZBnUiLINXlyQNeVSBcRiPD6y4GV7h0OBNvXuv1zFirLVkhQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1cc91e46e9e7faea712123f8d3e0ab7df9dd55ad79b181ae036a0643d726697","last_reissued_at":"2026-07-05T11:52:38.142108Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:52:38.142108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unconditional results for Artin-type problems over number fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pietro Sgobba","submitted_at":"2025-08-12T15:03:43Z","abstract_excerpt":"Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\\times$. For all but finitely many primes $\\mathfrak p$ of $K$, the reduction $(G \\bmod \\mathfrak p)$ generates a well-defined subgroup of the multiplicative group of the residue field at $\\mathfrak p$, and we may consider its index. We study the primes of $K$ for which this index lies in a given set of positive integers $S$. In particular, we prove that under certain convergence conditions on series associated to $S$ this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.08996","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.08996/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2508.08996","created_at":"2026-07-05T11:52:38.142175+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.08996v1","created_at":"2026-07-05T11:52:38.142175+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.08996","created_at":"2026-07-05T11:52:38.142175+00:00"},{"alias_kind":"pith_short_12","alias_value":"4HGJDZDOTZ72","created_at":"2026-07-05T11:52:38.142175+00:00"},{"alias_kind":"pith_short_16","alias_value":"4HGJDZDOTZ725JYS","created_at":"2026-07-05T11:52:38.142175+00:00"},{"alias_kind":"pith_short_8","alias_value":"4HGJDZDO","created_at":"2026-07-05T11:52:38.142175+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7","json":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7.json","graph_json":"https://pith.science/api/pith-number/4HGJDZDOTZ725JYSCI7Y2PQKW7/graph.json","events_json":"https://pith.science/api/pith-number/4HGJDZDOTZ725JYSCI7Y2PQKW7/events.json","paper":"https://pith.science/paper/4HGJDZDO"},"agent_actions":{"view_html":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7","download_json":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7.json","view_paper":"https://pith.science/paper/4HGJDZDO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.08996&json=true","fetch_graph":"https://pith.science/api/pith-number/4HGJDZDOTZ725JYSCI7Y2PQKW7/graph.json","fetch_events":"https://pith.science/api/pith-number/4HGJDZDOTZ725JYSCI7Y2PQKW7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7/action/storage_attestation","attest_author":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7/action/author_attestation","sign_citation":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7/action/citation_signature","submit_replication":"https://pith.science/pith/4HGJDZDOTZ725JYSCI7Y2PQKW7/action/replication_record"}},"created_at":"2026-07-05T11:52:38.142175+00:00","updated_at":"2026-07-05T11:52:38.142175+00:00"}