{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:L3QPMFK3VQIO4PTE3VYSFEUVZU","short_pith_number":"pith:L3QPMFK3","canonical_record":{"source":{"id":"1507.03431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-13T12:54:30Z","cross_cats_sorted":[],"title_canon_sha256":"9068f1ab187b27544310bc03b17e112cf14185f84641962173250eced4a05018","abstract_canon_sha256":"60c39dc0547aee9d8d18d771dd5a5342a0f88c00d9eb981475f7a55507557abf"},"schema_version":"1.0"},"canonical_sha256":"5ee0f6155bac10ee3e64dd71229295cd2c3c1a7c0e188babdbbedb1b64f34e44","source":{"kind":"arxiv","id":"1507.03431","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03431","created_at":"2026-05-18T01:36:58Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03431v1","created_at":"2026-05-18T01:36:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03431","created_at":"2026-05-18T01:36:58Z"},{"alias_kind":"pith_short_12","alias_value":"L3QPMFK3VQIO","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"L3QPMFK3VQIO4PTE","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"L3QPMFK3","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:L3QPMFK3VQIO4PTE3VYSFEUVZU","target":"record","payload":{"canonical_record":{"source":{"id":"1507.03431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-13T12:54:30Z","cross_cats_sorted":[],"title_canon_sha256":"9068f1ab187b27544310bc03b17e112cf14185f84641962173250eced4a05018","abstract_canon_sha256":"60c39dc0547aee9d8d18d771dd5a5342a0f88c00d9eb981475f7a55507557abf"},"schema_version":"1.0"},"canonical_sha256":"5ee0f6155bac10ee3e64dd71229295cd2c3c1a7c0e188babdbbedb1b64f34e44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:58.845025Z","signature_b64":"KUp7udko/IMoPYIEFb2O7TYLSYPxKLyzVjCHb5k2icwnC7vkfJHqwJQ0H3nihBbmEGzmguvPWt2JDiAJVTajCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ee0f6155bac10ee3e64dd71229295cd2c3c1a7c0e188babdbbedb1b64f34e44","last_reissued_at":"2026-05-18T01:36:58.844402Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:58.844402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.03431","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":"2026-05-18T01:36:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kwcf0S43e5x30r6PF3FWewGiE5NKr+NlKGsaw3yK915KpSCuz72nC5S10l2TjnlQogQ3yeGRkODfY29skMLCCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T15:13:34.251468Z"},"content_sha256":"52fef3d62671cfc79a89f62d8b580075e94b32d57d7f89c299ef11cc64f22c17","schema_version":"1.0","event_id":"sha256:52fef3d62671cfc79a89f62d8b580075e94b32d57d7f89c299ef11cc64f22c17"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:L3QPMFK3VQIO4PTE3VYSFEUVZU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the zeros of Dirichlet $L$-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kamel Mazhouda, Raouf Ouni, Sami Omar","submitted_at":"2015-07-13T12:54:30Z","abstract_excerpt":"In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010) no.4, 1109-1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03431","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":"2026-05-18T01:36:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jyelZkWnep1q8W/5joOMuS7fGUFi1WOOEAeKZ7dsYRyrUbTeUYp16GArTJ/Bd4tTkcFEogz2/Fct4yliQ2bKAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T15:13:34.251806Z"},"content_sha256":"2278d5af2118625881062bc3d4ee2d484686cb72bc3b541648fa6f339b4d2523","schema_version":"1.0","event_id":"sha256:2278d5af2118625881062bc3d4ee2d484686cb72bc3b541648fa6f339b4d2523"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU/bundle.json","state_url":"https://pith.science/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU/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":"2026-06-26T15:13:34Z","links":{"resolver":"https://pith.science/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU","bundle":"https://pith.science/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU/bundle.json","state":"https://pith.science/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L3QPMFK3VQIO4PTE3VYSFEUVZU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:L3QPMFK3VQIO4PTE3VYSFEUVZU","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":"60c39dc0547aee9d8d18d771dd5a5342a0f88c00d9eb981475f7a55507557abf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-13T12:54:30Z","title_canon_sha256":"9068f1ab187b27544310bc03b17e112cf14185f84641962173250eced4a05018"},"schema_version":"1.0","source":{"id":"1507.03431","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03431","created_at":"2026-05-18T01:36:58Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03431v1","created_at":"2026-05-18T01:36:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03431","created_at":"2026-05-18T01:36:58Z"},{"alias_kind":"pith_short_12","alias_value":"L3QPMFK3VQIO","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"L3QPMFK3VQIO4PTE","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"L3QPMFK3","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:2278d5af2118625881062bc3d4ee2d484686cb72bc3b541648fa6f339b4d2523","target":"graph","created_at":"2026-05-18T01:36:58Z","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 compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010) no.4, 1109-1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.","authors_text":"Kamel Mazhouda, Raouf Ouni, Sami Omar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-13T12:54:30Z","title":"On the zeros of Dirichlet $L$-functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03431","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":"sha256:52fef3d62671cfc79a89f62d8b580075e94b32d57d7f89c299ef11cc64f22c17","target":"record","created_at":"2026-05-18T01:36:58Z","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":"60c39dc0547aee9d8d18d771dd5a5342a0f88c00d9eb981475f7a55507557abf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-13T12:54:30Z","title_canon_sha256":"9068f1ab187b27544310bc03b17e112cf14185f84641962173250eced4a05018"},"schema_version":"1.0","source":{"id":"1507.03431","kind":"arxiv","version":1}},"canonical_sha256":"5ee0f6155bac10ee3e64dd71229295cd2c3c1a7c0e188babdbbedb1b64f34e44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ee0f6155bac10ee3e64dd71229295cd2c3c1a7c0e188babdbbedb1b64f34e44","first_computed_at":"2026-05-18T01:36:58.844402Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:58.844402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KUp7udko/IMoPYIEFb2O7TYLSYPxKLyzVjCHb5k2icwnC7vkfJHqwJQ0H3nihBbmEGzmguvPWt2JDiAJVTajCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:58.845025Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.03431","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52fef3d62671cfc79a89f62d8b580075e94b32d57d7f89c299ef11cc64f22c17","sha256:2278d5af2118625881062bc3d4ee2d484686cb72bc3b541648fa6f339b4d2523"],"state_sha256":"2b22edbaf9881c7ac82176c43ff9669b991da7dcec8d52f58ee46274fe79ff51"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T7H6QaIVBNtqzKbXA/MEPGdWvuw3knx4EiEt2Q614+YvY7naetWiys18TlW/hYA1Yc3B1OBF/q94Fl/yECoKCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T15:13:34.253765Z","bundle_sha256":"15ca0b9328a980185ada84f70b2c02e055329f374a7d43ff21dea98c267a1ef2"}}