{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:D3ZNF73FTFQ3EGVSJZDB4BR7ZU","short_pith_number":"pith:D3ZNF73F","canonical_record":{"source":{"id":"math/0612348","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-12-13T11:08:13Z","cross_cats_sorted":[],"title_canon_sha256":"019308e2892d1286e648c12aa1edd69f166a71c44e9b29060907452721d6fbab","abstract_canon_sha256":"f864d4568cbe1a8631cdf6b293b32acf91a915ba0874ac3ff4f8797491ff5a74"},"schema_version":"1.0"},"canonical_sha256":"1ef2d2ff659961b21ab24e461e063fcd2ecacbf60d630c371cbc1674b97f3591","source":{"kind":"arxiv","id":"math/0612348","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0612348","created_at":"2026-05-18T03:41:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0612348v2","created_at":"2026-05-18T03:41:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612348","created_at":"2026-05-18T03:41:31Z"},{"alias_kind":"pith_short_12","alias_value":"D3ZNF73FTFQ3","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"D3ZNF73FTFQ3EGVS","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"D3ZNF73F","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:D3ZNF73FTFQ3EGVSJZDB4BR7ZU","target":"record","payload":{"canonical_record":{"source":{"id":"math/0612348","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-12-13T11:08:13Z","cross_cats_sorted":[],"title_canon_sha256":"019308e2892d1286e648c12aa1edd69f166a71c44e9b29060907452721d6fbab","abstract_canon_sha256":"f864d4568cbe1a8631cdf6b293b32acf91a915ba0874ac3ff4f8797491ff5a74"},"schema_version":"1.0"},"canonical_sha256":"1ef2d2ff659961b21ab24e461e063fcd2ecacbf60d630c371cbc1674b97f3591","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:31.073786Z","signature_b64":"pY2uTYhBPbXEXxVY10uloWhy4bMvVQzrSpy5o0R6hZrLRClWkB5bcqcy+46ZyDcgAmiSsNIqeRatgApVDa73Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ef2d2ff659961b21ab24e461e063fcd2ecacbf60d630c371cbc1674b97f3591","last_reissued_at":"2026-05-18T03:41:31.073093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:31.073093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0612348","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e7KiqsyAF3+GDHNT5avgdUFO8wf4qv7x5VvXJwK/5f5TgCJiIKqG4LE8oVJr0I1fB8vK3eNhgR2mLMy5CjmICQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T12:53:47.586919Z"},"content_sha256":"295b2cdc7f15118e8672a14cc5282b485466c758ab79d13d97dd0a6eef150b8a","schema_version":"1.0","event_id":"sha256:295b2cdc7f15118e8672a14cc5282b485466c758ab79d13d97dd0a6eef150b8a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:D3ZNF73FTFQ3EGVSJZDB4BR7ZU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the mean values of L-functions in orthogonal and symplectic families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"H. M. Bui, J. P. Keating","submitted_at":"2006-12-13T11:08:13Z","abstract_excerpt":"Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612348","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HEiVSLtGKB4ntTLirQbQ7MDfHAIm9tYcj2t1VNaFh0ZXux/EMqw85NY8e/Mp5tSdb4YC7rp+lB8Q5sY8MuE7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T12:53:47.587256Z"},"content_sha256":"6b4deeac0cdb66f46a200a18a7e07d9151b6edcdad763b1982d1f3faebfeaaab","schema_version":"1.0","event_id":"sha256:6b4deeac0cdb66f46a200a18a7e07d9151b6edcdad763b1982d1f3faebfeaaab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU/bundle.json","state_url":"https://pith.science/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T12:53:47Z","links":{"resolver":"https://pith.science/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU","bundle":"https://pith.science/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU/bundle.json","state":"https://pith.science/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D3ZNF73FTFQ3EGVSJZDB4BR7ZU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:D3ZNF73FTFQ3EGVSJZDB4BR7ZU","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":"f864d4568cbe1a8631cdf6b293b32acf91a915ba0874ac3ff4f8797491ff5a74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-12-13T11:08:13Z","title_canon_sha256":"019308e2892d1286e648c12aa1edd69f166a71c44e9b29060907452721d6fbab"},"schema_version":"1.0","source":{"id":"math/0612348","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0612348","created_at":"2026-05-18T03:41:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0612348v2","created_at":"2026-05-18T03:41:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612348","created_at":"2026-05-18T03:41:31Z"},{"alias_kind":"pith_short_12","alias_value":"D3ZNF73FTFQ3","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"D3ZNF73FTFQ3EGVS","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"D3ZNF73F","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:6b4deeac0cdb66f46a200a18a7e07d9151b6edcdad763b1982d1f3faebfeaaab","target":"graph","created_at":"2026-05-18T03:41:31Z","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":"Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of ","authors_text":"H. M. Bui, J. P. Keating","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-12-13T11:08:13Z","title":"On the mean values of L-functions in orthogonal and symplectic families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612348","kind":"arxiv","version":2},"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":"sha256:295b2cdc7f15118e8672a14cc5282b485466c758ab79d13d97dd0a6eef150b8a","target":"record","created_at":"2026-05-18T03:41:31Z","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":"f864d4568cbe1a8631cdf6b293b32acf91a915ba0874ac3ff4f8797491ff5a74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-12-13T11:08:13Z","title_canon_sha256":"019308e2892d1286e648c12aa1edd69f166a71c44e9b29060907452721d6fbab"},"schema_version":"1.0","source":{"id":"math/0612348","kind":"arxiv","version":2}},"canonical_sha256":"1ef2d2ff659961b21ab24e461e063fcd2ecacbf60d630c371cbc1674b97f3591","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ef2d2ff659961b21ab24e461e063fcd2ecacbf60d630c371cbc1674b97f3591","first_computed_at":"2026-05-18T03:41:31.073093Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:31.073093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pY2uTYhBPbXEXxVY10uloWhy4bMvVQzrSpy5o0R6hZrLRClWkB5bcqcy+46ZyDcgAmiSsNIqeRatgApVDa73Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:31.073786Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0612348","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:295b2cdc7f15118e8672a14cc5282b485466c758ab79d13d97dd0a6eef150b8a","sha256:6b4deeac0cdb66f46a200a18a7e07d9151b6edcdad763b1982d1f3faebfeaaab"],"state_sha256":"af6e97c39a4666fc75d14b35ce51aa63f59fd1403f19209756425791fb9ababe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KP8sj6XOvEdc1EdiVIRGAVbmHmNABr1unMQXlWqDrzya047buW1cAcwrKe1AhIBI5FvPjWf/Srbrn54GFVCCBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T12:53:47.589187Z","bundle_sha256":"734706e1a6b9487661a26a46637dcde6de95787ac082e0cadd1f64df084280c0"}}