{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:KEYECEVPUI6LQ3O42GAYGQL5AI","short_pith_number":"pith:KEYECEVP","schema_version":"1.0","canonical_sha256":"51304112afa23cb86ddcd18183417d022090d4a8f5e7dcc38019c8157e1a282a","source":{"kind":"arxiv","id":"0907.0972","version":1},"attestation_state":"computed","paper":{"title":"On Witten multiple zeta-functions associated with semisimple Lie algebras IV","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hirofumi Tsumura, Kohji Matsumoto, Yasushi Komori","submitted_at":"2009-07-06T11:47:08Z","abstract_excerpt":"In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0907.0972","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-06T11:47:08Z","cross_cats_sorted":[],"title_canon_sha256":"bf5ae9d868f08564388fda2535524c228c6511dc3d779f5532081afae7f7ba74","abstract_canon_sha256":"c8627b88fda93fcb6dea20d967ad38298ba91a37a1e4016a4f3def35590ae1c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:07.765161Z","signature_b64":"yet3YrikppJfHwQzciYlR1lqMBAuGP4qD9+iYZ0r5vROOCDGM5DkUcjg4tVzvhT43rlCOZQygQcEPzt7/lIfBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51304112afa23cb86ddcd18183417d022090d4a8f5e7dcc38019c8157e1a282a","last_reissued_at":"2026-05-18T01:16:07.764622Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:07.764622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Witten multiple zeta-functions associated with semisimple Lie algebras IV","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hirofumi Tsumura, Kohji Matsumoto, Yasushi Komori","submitted_at":"2009-07-06T11:47:08Z","abstract_excerpt":"In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.0972","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0907.0972","created_at":"2026-05-18T01:16:07.764708+00:00"},{"alias_kind":"arxiv_version","alias_value":"0907.0972v1","created_at":"2026-05-18T01:16:07.764708+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.0972","created_at":"2026-05-18T01:16:07.764708+00:00"},{"alias_kind":"pith_short_12","alias_value":"KEYECEVPUI6L","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"KEYECEVPUI6LQ3O4","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"KEYECEVP","created_at":"2026-05-18T12:26:00.592388+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI","json":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI.json","graph_json":"https://pith.science/api/pith-number/KEYECEVPUI6LQ3O42GAYGQL5AI/graph.json","events_json":"https://pith.science/api/pith-number/KEYECEVPUI6LQ3O42GAYGQL5AI/events.json","paper":"https://pith.science/paper/KEYECEVP"},"agent_actions":{"view_html":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI","download_json":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI.json","view_paper":"https://pith.science/paper/KEYECEVP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0907.0972&json=true","fetch_graph":"https://pith.science/api/pith-number/KEYECEVPUI6LQ3O42GAYGQL5AI/graph.json","fetch_events":"https://pith.science/api/pith-number/KEYECEVPUI6LQ3O42GAYGQL5AI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI/action/storage_attestation","attest_author":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI/action/author_attestation","sign_citation":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI/action/citation_signature","submit_replication":"https://pith.science/pith/KEYECEVPUI6LQ3O42GAYGQL5AI/action/replication_record"}},"created_at":"2026-05-18T01:16:07.764708+00:00","updated_at":"2026-05-18T01:16:07.764708+00:00"}