{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FQS4RGSQQFW7N3VZHFJ3YM4ZRK","short_pith_number":"pith:FQS4RGSQ","canonical_record":{"source":{"id":"1602.00752","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-02-02T00:13:15Z","cross_cats_sorted":[],"title_canon_sha256":"c38a8fa47bccb2336689382f1cf722dbd31e7ad477754d6413dc98f803165456","abstract_canon_sha256":"904497c213538258556ad23fe7b1d76ecb68a9004669cf9778286a752fa1de28"},"schema_version":"1.0"},"canonical_sha256":"2c25c89a50816df6eeb93953bc33998a95c18f85ff5481950cdd779c0c816239","source":{"kind":"arxiv","id":"1602.00752","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00752","created_at":"2026-05-18T01:03:19Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00752v3","created_at":"2026-05-18T01:03:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00752","created_at":"2026-05-18T01:03:19Z"},{"alias_kind":"pith_short_12","alias_value":"FQS4RGSQQFW7","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FQS4RGSQQFW7N3VZ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FQS4RGSQ","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FQS4RGSQQFW7N3VZHFJ3YM4ZRK","target":"record","payload":{"canonical_record":{"source":{"id":"1602.00752","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-02-02T00:13:15Z","cross_cats_sorted":[],"title_canon_sha256":"c38a8fa47bccb2336689382f1cf722dbd31e7ad477754d6413dc98f803165456","abstract_canon_sha256":"904497c213538258556ad23fe7b1d76ecb68a9004669cf9778286a752fa1de28"},"schema_version":"1.0"},"canonical_sha256":"2c25c89a50816df6eeb93953bc33998a95c18f85ff5481950cdd779c0c816239","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:19.831046Z","signature_b64":"GFro/OaogDuamjpNc252b5YUQvthya4c6zriS/OsoJQPRuu5QTMWFIa2AeZwRtaE+q/n8TG1tqIdljuNXWXiDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c25c89a50816df6eeb93953bc33998a95c18f85ff5481950cdd779c0c816239","last_reissued_at":"2026-05-18T01:03:19.830452Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:19.830452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.00752","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LGU355He8hu8KYchFKMUiMfAxYKjwTYf+RzRvpZQon1EYzyrRuQF7kOLP7GBWfvLP8cv9TiCRfaqPXtnS/eDCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T06:24:55.071092Z"},"content_sha256":"03f9a8a478c689f5105abb393bedf7997c0dda29e7bff1157911ebe53e436934","schema_version":"1.0","event_id":"sha256:03f9a8a478c689f5105abb393bedf7997c0dda29e7bff1157911ebe53e436934"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FQS4RGSQQFW7N3VZHFJ3YM4ZRK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Zeta-polynomials for modular form periods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Sprung, Ken Ono, Larry Rolen","submitted_at":"2016-02-02T00:13:15Z","abstract_excerpt":"Answering problems of Manin, we use the critical $L$-values of even weight $k\\geq 4$ newforms $f\\in S_k(\\Gamma_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(\\rho)=0$, then $\\operatorname{Re}(\\rho)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and \"weighted moments\" of critical values $L$-values. In analogy with Ehr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00752","kind":"arxiv","version":3},"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":"2026-05-18T01:03:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vk/VZFhQQOYYNlEAdL2zuDkfpeC8FETT1KaQv3rTzRuKxs+jKgcdecFRBvEFMjb6DGQi1bjFs6S3/gCUZ+lUDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T06:24:55.071748Z"},"content_sha256":"e90ceb802705abceafb7f4c0ea3a244d038ce2385661252e741da6a4456dae47","schema_version":"1.0","event_id":"sha256:e90ceb802705abceafb7f4c0ea3a244d038ce2385661252e741da6a4456dae47"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK/bundle.json","state_url":"https://pith.science/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK/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":"2026-06-10T06:24:55Z","links":{"resolver":"https://pith.science/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK","bundle":"https://pith.science/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK/bundle.json","state":"https://pith.science/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FQS4RGSQQFW7N3VZHFJ3YM4ZRK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FQS4RGSQQFW7N3VZHFJ3YM4ZRK","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":"904497c213538258556ad23fe7b1d76ecb68a9004669cf9778286a752fa1de28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-02-02T00:13:15Z","title_canon_sha256":"c38a8fa47bccb2336689382f1cf722dbd31e7ad477754d6413dc98f803165456"},"schema_version":"1.0","source":{"id":"1602.00752","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00752","created_at":"2026-05-18T01:03:19Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00752v3","created_at":"2026-05-18T01:03:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00752","created_at":"2026-05-18T01:03:19Z"},{"alias_kind":"pith_short_12","alias_value":"FQS4RGSQQFW7","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FQS4RGSQQFW7N3VZ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FQS4RGSQ","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:e90ceb802705abceafb7f4c0ea3a244d038ce2385661252e741da6a4456dae47","target":"graph","created_at":"2026-05-18T01:03:19Z","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":"Answering problems of Manin, we use the critical $L$-values of even weight $k\\geq 4$ newforms $f\\in S_k(\\Gamma_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(\\rho)=0$, then $\\operatorname{Re}(\\rho)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and \"weighted moments\" of critical values $L$-values. In analogy with Ehr","authors_text":"Florian Sprung, Ken Ono, Larry Rolen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-02-02T00:13:15Z","title":"Zeta-polynomials for modular form periods"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00752","kind":"arxiv","version":3},"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":"sha256:03f9a8a478c689f5105abb393bedf7997c0dda29e7bff1157911ebe53e436934","target":"record","created_at":"2026-05-18T01:03:19Z","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":"904497c213538258556ad23fe7b1d76ecb68a9004669cf9778286a752fa1de28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-02-02T00:13:15Z","title_canon_sha256":"c38a8fa47bccb2336689382f1cf722dbd31e7ad477754d6413dc98f803165456"},"schema_version":"1.0","source":{"id":"1602.00752","kind":"arxiv","version":3}},"canonical_sha256":"2c25c89a50816df6eeb93953bc33998a95c18f85ff5481950cdd779c0c816239","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c25c89a50816df6eeb93953bc33998a95c18f85ff5481950cdd779c0c816239","first_computed_at":"2026-05-18T01:03:19.830452Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:19.830452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GFro/OaogDuamjpNc252b5YUQvthya4c6zriS/OsoJQPRuu5QTMWFIa2AeZwRtaE+q/n8TG1tqIdljuNXWXiDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:19.831046Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.00752","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03f9a8a478c689f5105abb393bedf7997c0dda29e7bff1157911ebe53e436934","sha256:e90ceb802705abceafb7f4c0ea3a244d038ce2385661252e741da6a4456dae47"],"state_sha256":"739c517dc7b91ac951f9173a6bd5e9f60f03e8fa0e2c47533200d37a12e0b726"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/NRq5LJRCfcAQbHWwcm4PUkBU78NQedOHdVCiorgKiaJ9jH8LVrwYB0YiCJu6MGeZj1Rm5hhX9KrSP81bgg6CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T06:24:55.075534Z","bundle_sha256":"7636329c638b71a2e23ce174c9fe89771a848fb84b48a26a72d9a85b71a196be"}}