{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ETKBL5KQOFDITISOD3ZSHUIYHJ","short_pith_number":"pith:ETKBL5KQ","canonical_record":{"source":{"id":"1402.1585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-07T10:10:56Z","cross_cats_sorted":[],"title_canon_sha256":"511019c95791ce0041a6de03050d2fc668234b57bfbb16000b3d0a9eb4aea7bb","abstract_canon_sha256":"b2913f7d7ac30ddb99c06026ba1fd6e9a772496811386e1472c839ff46d13e85"},"schema_version":"1.0"},"canonical_sha256":"24d415f550714689a24e1ef323d1183a48a02e82b2befb739f57e1d6b1222628","source":{"kind":"arxiv","id":"1402.1585","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.1585","created_at":"2026-05-18T02:59:46Z"},{"alias_kind":"arxiv_version","alias_value":"1402.1585v1","created_at":"2026-05-18T02:59:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.1585","created_at":"2026-05-18T02:59:46Z"},{"alias_kind":"pith_short_12","alias_value":"ETKBL5KQOFDI","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"ETKBL5KQOFDITISO","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"ETKBL5KQ","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ETKBL5KQOFDITISOD3ZSHUIYHJ","target":"record","payload":{"canonical_record":{"source":{"id":"1402.1585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-07T10:10:56Z","cross_cats_sorted":[],"title_canon_sha256":"511019c95791ce0041a6de03050d2fc668234b57bfbb16000b3d0a9eb4aea7bb","abstract_canon_sha256":"b2913f7d7ac30ddb99c06026ba1fd6e9a772496811386e1472c839ff46d13e85"},"schema_version":"1.0"},"canonical_sha256":"24d415f550714689a24e1ef323d1183a48a02e82b2befb739f57e1d6b1222628","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:46.792219Z","signature_b64":"VkzaM35Nfh3jCzyY5N3HicjXZIxOo+saLeTtzSCQ3jXDIRok1+iJtHxKjU3OXpdn8kN6w/GVY7262tVnEPmOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24d415f550714689a24e1ef323d1183a48a02e82b2befb739f57e1d6b1222628","last_reissued_at":"2026-05-18T02:59:46.791277Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:46.791277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.1585","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:59:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1KdGIiVhRHRCZc0c8P17MX3MTzx7puPzWNIdqTkaMHnMRx8IPZkejw2VPGP5yMqTOmqQ2ru4UYUkBL5Qq7+lBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T02:29:36.277944Z"},"content_sha256":"7d0ebfbfe07dd93fa3ec49ab6cf77949f40ac352e65782bf9089c5914dedb399","schema_version":"1.0","event_id":"sha256:7d0ebfbfe07dd93fa3ec49ab6cf77949f40ac352e65782bf9089c5914dedb399"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ETKBL5KQOFDITISOD3ZSHUIYHJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eisenstein series identities based on partial fraction decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Koji Tasaka, Minoru Hirose, Nobuo Sato","submitted_at":"2014-02-07T10:10:56Z","abstract_excerpt":"From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\\ (2\\le i \\le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\\rm SL}_2({\\mathbb Z})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1585","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":""},"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":"2026-05-18T02:59:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aPDVPaGtnNLk6ywNZmeK12j/U3WpsOhfKZ/cqsNmVpjfCgliiJm/jQe8AV6ohrbDkgl+ienWkPtGjNXCBXVwCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T02:29:36.278600Z"},"content_sha256":"dd51d43983bb3dbf15a52b934fc535c1c6eb643e8b3922fc7f3fc32ad72d662e","schema_version":"1.0","event_id":"sha256:dd51d43983bb3dbf15a52b934fc535c1c6eb643e8b3922fc7f3fc32ad72d662e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ/bundle.json","state_url":"https://pith.science/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ/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":"2026-05-27T02:29:36Z","links":{"resolver":"https://pith.science/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ","bundle":"https://pith.science/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ/bundle.json","state":"https://pith.science/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ETKBL5KQOFDITISOD3ZSHUIYHJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ETKBL5KQOFDITISOD3ZSHUIYHJ","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":"b2913f7d7ac30ddb99c06026ba1fd6e9a772496811386e1472c839ff46d13e85","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-07T10:10:56Z","title_canon_sha256":"511019c95791ce0041a6de03050d2fc668234b57bfbb16000b3d0a9eb4aea7bb"},"schema_version":"1.0","source":{"id":"1402.1585","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.1585","created_at":"2026-05-18T02:59:46Z"},{"alias_kind":"arxiv_version","alias_value":"1402.1585v1","created_at":"2026-05-18T02:59:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.1585","created_at":"2026-05-18T02:59:46Z"},{"alias_kind":"pith_short_12","alias_value":"ETKBL5KQOFDI","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"ETKBL5KQOFDITISO","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"ETKBL5KQ","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:dd51d43983bb3dbf15a52b934fc535c1c6eb643e8b3922fc7f3fc32ad72d662e","target":"graph","created_at":"2026-05-18T02:59:46Z","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":"From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\\ (2\\le i \\le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\\rm SL}_2({\\mathbb Z})$.","authors_text":"Koji Tasaka, Minoru Hirose, Nobuo Sato","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-07T10:10:56Z","title":"Eisenstein series identities based on partial fraction decomposition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1585","kind":"arxiv","version":1},"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":"sha256:7d0ebfbfe07dd93fa3ec49ab6cf77949f40ac352e65782bf9089c5914dedb399","target":"record","created_at":"2026-05-18T02:59:46Z","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":"b2913f7d7ac30ddb99c06026ba1fd6e9a772496811386e1472c839ff46d13e85","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-07T10:10:56Z","title_canon_sha256":"511019c95791ce0041a6de03050d2fc668234b57bfbb16000b3d0a9eb4aea7bb"},"schema_version":"1.0","source":{"id":"1402.1585","kind":"arxiv","version":1}},"canonical_sha256":"24d415f550714689a24e1ef323d1183a48a02e82b2befb739f57e1d6b1222628","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24d415f550714689a24e1ef323d1183a48a02e82b2befb739f57e1d6b1222628","first_computed_at":"2026-05-18T02:59:46.791277Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:59:46.791277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VkzaM35Nfh3jCzyY5N3HicjXZIxOo+saLeTtzSCQ3jXDIRok1+iJtHxKjU3OXpdn8kN6w/GVY7262tVnEPmOCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:59:46.792219Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.1585","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d0ebfbfe07dd93fa3ec49ab6cf77949f40ac352e65782bf9089c5914dedb399","sha256:dd51d43983bb3dbf15a52b934fc535c1c6eb643e8b3922fc7f3fc32ad72d662e"],"state_sha256":"97d046054439eed3f6bd51059354d34cbe8306789b7475ea0886d2c098dd62ac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lMKv9xs8u+BmvyBim0LBOs0Z11oROUf0e7vCs42qHlSHFFmXq+2VgQMgfX5amA5JPUhQysR9ooorWsOKHBPuBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T02:29:36.281812Z","bundle_sha256":"88d61b98c558dfd27816c3a3f18f7366815ed866bfa55ad892cfe29f9e49cdab"}}