{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:TGSELTGU4DSGFVIPWKA2O7543Q","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2fdeab69251bf88979d8f6fbe11224ee7db49e7dfbd039d63ff09d4a0bd7a430","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2006-02-07T01:52:33Z","title_canon_sha256":"98a7fc2235963df992ec07d7d73f61edafc9c00ff65b8105faf6380606b5789f"},"schema_version":"1.0","source":{"id":"math/0602116","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602116","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602116v4","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602116","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"TGSELTGU4DSG","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"TGSELTGU4DSGFVIP","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"TGSELTGU","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:bace6a00bb0494340656029caec6c6ec9827216b44579c1f6cbe177da37de414","target":"graph","created_at":"2026-05-18T01:38:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form $p=am^2+1$ such that $a\\leq p^{5/9+\\epsilon}$.","authors_text":"Liangyi Zhao, Stephan Baier","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2006-02-07T01:52:33Z","title":"Bombieri-Vinogradov Type Theorem for Sparse Sets of Moduli"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602116","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ce06a8cab1bc72162e2d0d5dc0f4046803f6110619b9cfeb8bb33cfba3fa608e","target":"record","created_at":"2026-05-18T01:38:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2fdeab69251bf88979d8f6fbe11224ee7db49e7dfbd039d63ff09d4a0bd7a430","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2006-02-07T01:52:33Z","title_canon_sha256":"98a7fc2235963df992ec07d7d73f61edafc9c00ff65b8105faf6380606b5789f"},"schema_version":"1.0","source":{"id":"math/0602116","kind":"arxiv","version":4}},"canonical_sha256":"99a445ccd4e0e462d50fb281a77fbcdc38eda826984e5a9e710c05c8c4b0b197","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99a445ccd4e0e462d50fb281a77fbcdc38eda826984e5a9e710c05c8c4b0b197","first_computed_at":"2026-05-18T01:38:24.417277Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:24.417277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TwnIAORc/fi+R6ua2AiLJM9Rjzgo655RVVQHhCyHMgKejMXi32leXJidOAmCIfM4HWIdpZNq3j+vjBx18M9xCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:24.418053Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0602116","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce06a8cab1bc72162e2d0d5dc0f4046803f6110619b9cfeb8bb33cfba3fa608e","sha256:bace6a00bb0494340656029caec6c6ec9827216b44579c1f6cbe177da37de414"],"state_sha256":"071798db9684265b02fb9f0586d02b99d2f62526049b33d8b90c5f155fab5e10"}