{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:UCEN6TQVLMYNKRRJNOJTZZ244H","short_pith_number":"pith:UCEN6TQV","canonical_record":{"source":{"id":"math/0604312","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2006-04-13T14:04:02Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"18c314011ebda2feb92cafb1ee91d6d96f729e9ada85dfd634f407d68f207ea3","abstract_canon_sha256":"1282f48d2de9baf7804923f03937004f343cb1e0709b1cff14f175ec4b3967af"},"schema_version":"1.0"},"canonical_sha256":"a088df4e155b30d546296b933ce75ce1db322c0043e3b61018a74016974d34cd","source":{"kind":"arxiv","id":"math/0604312","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0604312","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0604312v1","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0604312","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"pith_short_12","alias_value":"UCEN6TQVLMYN","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"UCEN6TQVLMYNKRRJ","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"UCEN6TQV","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:UCEN6TQVLMYNKRRJNOJTZZ244H","target":"record","payload":{"canonical_record":{"source":{"id":"math/0604312","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2006-04-13T14:04:02Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"18c314011ebda2feb92cafb1ee91d6d96f729e9ada85dfd634f407d68f207ea3","abstract_canon_sha256":"1282f48d2de9baf7804923f03937004f343cb1e0709b1cff14f175ec4b3967af"},"schema_version":"1.0"},"canonical_sha256":"a088df4e155b30d546296b933ce75ce1db322c0043e3b61018a74016974d34cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:24.441924Z","signature_b64":"NYhIw4X14zulISriBHNmh5umDuFzwZpf6xCjAOP6pbjn6nCqAMFWoz05m1QjSycZf9N0DfyiYVHUGUCQpmYNCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a088df4e155b30d546296b933ce75ce1db322c0043e3b61018a74016974d34cd","last_reissued_at":"2026-05-18T03:11:24.441188Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:24.441188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0604312","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":"2026-05-18T03:11:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PHlafBgkht9UeukYab/4900AgfuOts1Ek6G0Gpo/W9VStC/yMv1X5KBSiEWO6TFcJPk2gjpFROAb8Pe66RjkCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T04:31:29.450010Z"},"content_sha256":"7b163b9d057591681ee99f2f7e16a9ec5302907ab57abc5ae535ff56adc88b20","schema_version":"1.0","event_id":"sha256:7b163b9d057591681ee99f2f7e16a9ec5302907ab57abc5ae535ff56adc88b20"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:UCEN6TQVLMYNKRRJNOJTZZ244H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Irrationality of $\\zeta_q(1)$ and $\\zeta_q(2)$","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Kelly Postelmans, Walter Van Assche","submitted_at":"2006-04-13T14:04:02Z","abstract_excerpt":"In this paper we show how one can obtain simultaneous rational approximants for $\\zeta_q(1)$ and $\\zeta_q(2)$ with a common denominator by means of Hermite-Pade approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, $\\zeta_q(1)$, $\\zeta_q(2)$ are linearly independent over the rationals. In particular this implies that $\\zeta_q(1)$ and $\\zeta_q(2)$ are irrational. Furthermore we give an upper bound for the measure of irrationality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0604312","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":"2026-05-18T03:11:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H/4C4gcHspFtDZ/qhFFkiLYq4LakIMwDnMBTiJosmtVMWwmDtqMVRaooCVS+8XR3wT8ncxDAITrQG7ivxIKuDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T04:31:29.450700Z"},"content_sha256":"94befe8eef01f200a85968af4bd6f8e9058c9529828b5a7d41f3e14fd30e1357","schema_version":"1.0","event_id":"sha256:94befe8eef01f200a85968af4bd6f8e9058c9529828b5a7d41f3e14fd30e1357"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UCEN6TQVLMYNKRRJNOJTZZ244H/bundle.json","state_url":"https://pith.science/pith/UCEN6TQVLMYNKRRJNOJTZZ244H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UCEN6TQVLMYNKRRJNOJTZZ244H/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":"2026-06-05T04:31:29Z","links":{"resolver":"https://pith.science/pith/UCEN6TQVLMYNKRRJNOJTZZ244H","bundle":"https://pith.science/pith/UCEN6TQVLMYNKRRJNOJTZZ244H/bundle.json","state":"https://pith.science/pith/UCEN6TQVLMYNKRRJNOJTZZ244H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UCEN6TQVLMYNKRRJNOJTZZ244H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:UCEN6TQVLMYNKRRJNOJTZZ244H","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":"1282f48d2de9baf7804923f03937004f343cb1e0709b1cff14f175ec4b3967af","cross_cats_sorted":["math.NT"],"license":"","primary_cat":"math.CA","submitted_at":"2006-04-13T14:04:02Z","title_canon_sha256":"18c314011ebda2feb92cafb1ee91d6d96f729e9ada85dfd634f407d68f207ea3"},"schema_version":"1.0","source":{"id":"math/0604312","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0604312","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0604312v1","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0604312","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"pith_short_12","alias_value":"UCEN6TQVLMYN","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"UCEN6TQVLMYNKRRJ","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"UCEN6TQV","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:94befe8eef01f200a85968af4bd6f8e9058c9529828b5a7d41f3e14fd30e1357","target":"graph","created_at":"2026-05-18T03:11:24Z","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":"In this paper we show how one can obtain simultaneous rational approximants for $\\zeta_q(1)$ and $\\zeta_q(2)$ with a common denominator by means of Hermite-Pade approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, $\\zeta_q(1)$, $\\zeta_q(2)$ are linearly independent over the rationals. In particular this implies that $\\zeta_q(1)$ and $\\zeta_q(2)$ are irrational. Furthermore we give an upper bound for the measure of irrationality.","authors_text":"Kelly Postelmans, Walter Van Assche","cross_cats":["math.NT"],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2006-04-13T14:04:02Z","title":"Irrationality of $\\zeta_q(1)$ and $\\zeta_q(2)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0604312","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":"sha256:7b163b9d057591681ee99f2f7e16a9ec5302907ab57abc5ae535ff56adc88b20","target":"record","created_at":"2026-05-18T03:11:24Z","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":"1282f48d2de9baf7804923f03937004f343cb1e0709b1cff14f175ec4b3967af","cross_cats_sorted":["math.NT"],"license":"","primary_cat":"math.CA","submitted_at":"2006-04-13T14:04:02Z","title_canon_sha256":"18c314011ebda2feb92cafb1ee91d6d96f729e9ada85dfd634f407d68f207ea3"},"schema_version":"1.0","source":{"id":"math/0604312","kind":"arxiv","version":1}},"canonical_sha256":"a088df4e155b30d546296b933ce75ce1db322c0043e3b61018a74016974d34cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a088df4e155b30d546296b933ce75ce1db322c0043e3b61018a74016974d34cd","first_computed_at":"2026-05-18T03:11:24.441188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:24.441188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NYhIw4X14zulISriBHNmh5umDuFzwZpf6xCjAOP6pbjn6nCqAMFWoz05m1QjSycZf9N0DfyiYVHUGUCQpmYNCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:24.441924Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0604312","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7b163b9d057591681ee99f2f7e16a9ec5302907ab57abc5ae535ff56adc88b20","sha256:94befe8eef01f200a85968af4bd6f8e9058c9529828b5a7d41f3e14fd30e1357"],"state_sha256":"15a37feb3672eaf0c5867bd2ba68471e6d372e77d38ae17c6e6fd2abaf7c355c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7EgqQe0DX6tkpRQyYUaqlhlW3SEPy/40j2/kgE1ExuJLpHgFyRp9kc76DSGqHTCrq5Pu+wr7G09sJxrsdf+uAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T04:31:29.454687Z","bundle_sha256":"3acc3603ae0b55567f167ee78ced1744bc6645d9020d58f4900cddf9ceb3a697"}}