{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:EHUHM4G3MUBUXJHFWQAMZLGWMM","short_pith_number":"pith:EHUHM4G3","canonical_record":{"source":{"id":"1802.01374","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T13:02:33Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"5d26a2a1dad30cec1b00c7776c355a7571d87a87bcc96359ba8ad4fe47b5dfa3","abstract_canon_sha256":"8d97ab9ce69e5032318aa553f41e7b6c76dcd88437b70dcc7af9d482d8737662"},"schema_version":"1.0"},"canonical_sha256":"21e87670db65034ba4e5b400ccacd663214e12674bf82c1fcbcd9ab97288669c","source":{"kind":"arxiv","id":"1802.01374","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.01374","created_at":"2026-05-18T00:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"1802.01374v2","created_at":"2026-05-18T00:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01374","created_at":"2026-05-18T00:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"EHUHM4G3MUBU","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EHUHM4G3MUBUXJHF","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EHUHM4G3","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:EHUHM4G3MUBUXJHFWQAMZLGWMM","target":"record","payload":{"canonical_record":{"source":{"id":"1802.01374","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T13:02:33Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"5d26a2a1dad30cec1b00c7776c355a7571d87a87bcc96359ba8ad4fe47b5dfa3","abstract_canon_sha256":"8d97ab9ce69e5032318aa553f41e7b6c76dcd88437b70dcc7af9d482d8737662"},"schema_version":"1.0"},"canonical_sha256":"21e87670db65034ba4e5b400ccacd663214e12674bf82c1fcbcd9ab97288669c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:20.214151Z","signature_b64":"gIWV7O4ET4kG504i/hioEod9TOq3yaJ3gGqqQ/qWVZOuZKcHtLLh+/6mceLEQwXGvMGPHi3u4XJIno8iVbIxBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21e87670db65034ba4e5b400ccacd663214e12674bf82c1fcbcd9ab97288669c","last_reissued_at":"2026-05-18T00:21:20.213529Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:20.213529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.01374","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":"2026-05-18T00:21:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4wqAH7pQW7Ingm6EX1XqzfeEFnqxLiBkJjX3qarUyBwdRPLOPgVYw1BCd9bcNPgEptyVF7lgdg5J62VFYUmvBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T13:00:05.509173Z"},"content_sha256":"a72baf5478260c1f243a8ef3f06fd290ff10eba8b7023f73a6d5847b4b1b1e63","schema_version":"1.0","event_id":"sha256:a72baf5478260c1f243a8ef3f06fd290ff10eba8b7023f73a6d5847b4b1b1e63"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:EHUHM4G3MUBUXJHFWQAMZLGWMM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Congruences for the Coefficients of the Powers of the Euler Product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Edward Y.S. Liu, Jack C.D. Zhao, Julia Q.D. Du","submitted_at":"2018-02-05T13:02:33Z","abstract_excerpt":"Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\\prod _{n=1}^{\\infty}(1-q^n)^k=\\sum_{n=0}^{\\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\\alpha} n +\\frac{k(2^{2\\alpha}-1)}{3})$ $(1\\leq k\\leq 3)$ and $p_{3k} (3^{2\\beta}n+\\frac{k(3^{2\\beta}-1)}{8})$ $(1\\leq k\\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite famil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01374","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":"2026-05-18T00:21:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+XbEgiCUXi8BgamJvCm4/LO0adni8UxMyZbUHNYPjW56p5+01ZWOXGfrtDAzxdk3lr++yOa3fAVtK4xymFgQCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T13:00:05.509597Z"},"content_sha256":"29e153a5ed3e8a99c5f045e15e126821dd361ede1413aa8e28b4c1781da00542","schema_version":"1.0","event_id":"sha256:29e153a5ed3e8a99c5f045e15e126821dd361ede1413aa8e28b4c1781da00542"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/bundle.json","state_url":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/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":"2026-06-21T13:00:05Z","links":{"resolver":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM","bundle":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/bundle.json","state":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:EHUHM4G3MUBUXJHFWQAMZLGWMM","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":"8d97ab9ce69e5032318aa553f41e7b6c76dcd88437b70dcc7af9d482d8737662","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T13:02:33Z","title_canon_sha256":"5d26a2a1dad30cec1b00c7776c355a7571d87a87bcc96359ba8ad4fe47b5dfa3"},"schema_version":"1.0","source":{"id":"1802.01374","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.01374","created_at":"2026-05-18T00:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"1802.01374v2","created_at":"2026-05-18T00:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01374","created_at":"2026-05-18T00:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"EHUHM4G3MUBU","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EHUHM4G3MUBUXJHF","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EHUHM4G3","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:29e153a5ed3e8a99c5f045e15e126821dd361ede1413aa8e28b4c1781da00542","target":"graph","created_at":"2026-05-18T00:21:20Z","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":"Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\\prod _{n=1}^{\\infty}(1-q^n)^k=\\sum_{n=0}^{\\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\\alpha} n +\\frac{k(2^{2\\alpha}-1)}{3})$ $(1\\leq k\\leq 3)$ and $p_{3k} (3^{2\\beta}n+\\frac{k(3^{2\\beta}-1)}{8})$ $(1\\leq k\\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite famil","authors_text":"Edward Y.S. Liu, Jack C.D. Zhao, Julia Q.D. Du","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T13:02:33Z","title":"Congruences for the Coefficients of the Powers of the Euler Product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01374","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:a72baf5478260c1f243a8ef3f06fd290ff10eba8b7023f73a6d5847b4b1b1e63","target":"record","created_at":"2026-05-18T00:21:20Z","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":"8d97ab9ce69e5032318aa553f41e7b6c76dcd88437b70dcc7af9d482d8737662","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T13:02:33Z","title_canon_sha256":"5d26a2a1dad30cec1b00c7776c355a7571d87a87bcc96359ba8ad4fe47b5dfa3"},"schema_version":"1.0","source":{"id":"1802.01374","kind":"arxiv","version":2}},"canonical_sha256":"21e87670db65034ba4e5b400ccacd663214e12674bf82c1fcbcd9ab97288669c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21e87670db65034ba4e5b400ccacd663214e12674bf82c1fcbcd9ab97288669c","first_computed_at":"2026-05-18T00:21:20.213529Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:20.213529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gIWV7O4ET4kG504i/hioEod9TOq3yaJ3gGqqQ/qWVZOuZKcHtLLh+/6mceLEQwXGvMGPHi3u4XJIno8iVbIxBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:20.214151Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.01374","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a72baf5478260c1f243a8ef3f06fd290ff10eba8b7023f73a6d5847b4b1b1e63","sha256:29e153a5ed3e8a99c5f045e15e126821dd361ede1413aa8e28b4c1781da00542"],"state_sha256":"e230a055349ca256e6e0f24eb60946ebecfd82fa7cb11814eadbc0c0b8efd301"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9PT9C4yvzpsdOGxeX24XeIOFcYEY1NMtpQ4k/Pe7E7DO+EWqgvjewkJu/YFyXZms3JDt/Y0curpP/iRtY0U1Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T13:00:05.511625Z","bundle_sha256":"474e11c6eb3b188c08f06555cdeeb83407e1f400fc7a593aac6633f1e112dedd"}}