{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:6AKDJE6GO27KBG7P2BLR2Q2DEP","short_pith_number":"pith:6AKDJE6G","schema_version":"1.0","canonical_sha256":"f0143493c676bea09befd0571d434323cf8b2f7adf5f84a955b3f2b6521d7fbc","source":{"kind":"arxiv","id":"2201.12321","version":1},"attestation_state":"computed","paper":{"title":"Dimension Free Growth Results for Vector-Valued Functions of Several Complex Variables","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Faruk F. Abi-Khuzam","submitted_at":"2022-01-28T18:37:12Z","abstract_excerpt":"Let f be an entire function of finite order less than 1. The maximum modulus M(r) of f and the counting function of the zeros N(r) are connected by a best possible growth inequality known as Valiron's Theorem:\n  For functions subharmonic in d-dimensional Euclidean space, Hayman obtained a corresponding result with a best possible constant involving the dimension d. For the special case of an entire function on d-dimensional complex space, we obtain a corresponding dimension-free, best possible inequality."},"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":"2201.12321","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2022-01-28T18:37:12Z","cross_cats_sorted":[],"title_canon_sha256":"35947f7fcdcdb9c7dfb4f23352620fc5a233ee077cf170500b356f6792ce3587","abstract_canon_sha256":"a4bf0c336a34047000786b5f5801fa358df4333cde68b713059273615fe6fd01"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:52:22.025781Z","signature_b64":"rm/urpnI17Y+z2can4Ou/aRHMEHorpQY0ejI0mhAbnCZgGjpeQR9YDYELYp+G4lqGpOWPlfp9k7j0/aJyg7WBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0143493c676bea09befd0571d434323cf8b2f7adf5f84a955b3f2b6521d7fbc","last_reissued_at":"2026-07-05T03:52:22.025435Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:52:22.025435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dimension Free Growth Results for Vector-Valued Functions of Several Complex Variables","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Faruk F. Abi-Khuzam","submitted_at":"2022-01-28T18:37:12Z","abstract_excerpt":"Let f be an entire function of finite order less than 1. The maximum modulus M(r) of f and the counting function of the zeros N(r) are connected by a best possible growth inequality known as Valiron's Theorem:\n  For functions subharmonic in d-dimensional Euclidean space, Hayman obtained a corresponding result with a best possible constant involving the dimension d. For the special case of an entire function on d-dimensional complex space, we obtain a corresponding dimension-free, best possible inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.12321","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/2201.12321/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":"2201.12321","created_at":"2026-07-05T03:52:22.025490+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.12321v1","created_at":"2026-07-05T03:52:22.025490+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.12321","created_at":"2026-07-05T03:52:22.025490+00:00"},{"alias_kind":"pith_short_12","alias_value":"6AKDJE6GO27K","created_at":"2026-07-05T03:52:22.025490+00:00"},{"alias_kind":"pith_short_16","alias_value":"6AKDJE6GO27KBG7P","created_at":"2026-07-05T03:52:22.025490+00:00"},{"alias_kind":"pith_short_8","alias_value":"6AKDJE6G","created_at":"2026-07-05T03:52:22.025490+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2509.01259","citing_title":"ReCap: Event-Aware Image Captioning with Article Retrieval and Semantic Gaussian Normalization","ref_index":26,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP","json":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP.json","graph_json":"https://pith.science/api/pith-number/6AKDJE6GO27KBG7P2BLR2Q2DEP/graph.json","events_json":"https://pith.science/api/pith-number/6AKDJE6GO27KBG7P2BLR2Q2DEP/events.json","paper":"https://pith.science/paper/6AKDJE6G"},"agent_actions":{"view_html":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP","download_json":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP.json","view_paper":"https://pith.science/paper/6AKDJE6G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.12321&json=true","fetch_graph":"https://pith.science/api/pith-number/6AKDJE6GO27KBG7P2BLR2Q2DEP/graph.json","fetch_events":"https://pith.science/api/pith-number/6AKDJE6GO27KBG7P2BLR2Q2DEP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP/action/storage_attestation","attest_author":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP/action/author_attestation","sign_citation":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP/action/citation_signature","submit_replication":"https://pith.science/pith/6AKDJE6GO27KBG7P2BLR2Q2DEP/action/replication_record"}},"created_at":"2026-07-05T03:52:22.025490+00:00","updated_at":"2026-07-05T03:52:22.025490+00:00"}