{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:SFNMCQOJ6A3HGF55RX73IXS2MO","short_pith_number":"pith:SFNMCQOJ","canonical_record":{"source":{"id":"1708.02653","kind":"arxiv","version":22},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-08-07T12:53:52Z","cross_cats_sorted":[],"title_canon_sha256":"994a470dac3d3bb5f9e1d9bbc781cc4ec6d463f6d977f63c21bad67ad5b594c8","abstract_canon_sha256":"9474ba7237701445deb06e6732cce125cea1d312665e3817a1e8f60242f7cee7"},"schema_version":"1.0"},"canonical_sha256":"915ac141c9f0367317bd8dffb45e5a63a03160f43805176415495511df2e7b91","source":{"kind":"arxiv","id":"1708.02653","version":22},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.02653","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"1708.02653v22","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.02653","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"SFNMCQOJ6A3H","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"pith_short_16","alias_value":"SFNMCQOJ6A3HGF55","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"pith_short_8","alias_value":"SFNMCQOJ","created_at":"2026-06-02T03:04:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:SFNMCQOJ6A3HGF55RX73IXS2MO","target":"record","payload":{"canonical_record":{"source":{"id":"1708.02653","kind":"arxiv","version":22},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-08-07T12:53:52Z","cross_cats_sorted":[],"title_canon_sha256":"994a470dac3d3bb5f9e1d9bbc781cc4ec6d463f6d977f63c21bad67ad5b594c8","abstract_canon_sha256":"9474ba7237701445deb06e6732cce125cea1d312665e3817a1e8f60242f7cee7"},"schema_version":"1.0"},"canonical_sha256":"915ac141c9f0367317bd8dffb45e5a63a03160f43805176415495511df2e7b91","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T03:04:29.525449Z","signature_b64":"EnutID6ny0j8bromf9+yRepNzxST374QQ7UNyjx7lXOc7uQcCgPjXqNQ/fIoAjv+9Im+xZ6P9ykg1q30V0tWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"915ac141c9f0367317bd8dffb45e5a63a03160f43805176415495511df2e7b91","last_reissued_at":"2026-06-02T03:04:29.524830Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T03:04:29.524830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.02653","source_version":22,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T03:04:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+IZvU1cyOgZlKkMjvMeb5vnSfk3mfAFJFWTTbWoIIfRdxVhkqIW9Za6Z4kbh6JWQL4dPOP4lCc8PlAPbpOUSAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:33:47.349698Z"},"content_sha256":"f293cee47e1f9777916c0c6f094cc03d903a273de517a26c75f5fc75c103e28a","schema_version":"1.0","event_id":"sha256:f293cee47e1f9777916c0c6f094cc03d903a273de517a26c75f5fc75c103e28a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:SFNMCQOJ6A3HGF55RX73IXS2MO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Hilbert's 8th Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Nicholas G. Polson","submitted_at":"2017-08-07T12:53:52Z","abstract_excerpt":"A Hadamard factorisation of the Riemann $\\xi$-function is constructed to characterise the zeros of the Riemann zeta function. The argument proceeds via a probabilistic reformulation: the Riemann Hypothesis is shown to be equivalent to Thorin's condition, namely that $\\xi(\\half)/\\xi(\\half+\\sqrt s\\,)$ is the Laplace transform of a generalised gamma convolution (GGC). A GGC random variable realising this Laplace transform is constructed unconditionally, using L\\'evy representations of the Gamma function, the Euler product for the zeta function, and Ramanujan's Master Theorem, thereby establishing"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02653","kind":"arxiv","version":22},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1708.02653/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-02T03:04:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZEl7FWyvHxeR22rlmWagXZLdCUUj1Q+rWHAG33HZYLYpafyxuAQvOq5Fa8MjDIBDTNA6T0QtNZiMgEuEURyrAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:33:47.350548Z"},"content_sha256":"6797df73c9632ca34c3d86770dea21c64a171fbc0ab79323bf18fe378875d640","schema_version":"1.0","event_id":"sha256:6797df73c9632ca34c3d86770dea21c64a171fbc0ab79323bf18fe378875d640"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SFNMCQOJ6A3HGF55RX73IXS2MO/bundle.json","state_url":"https://pith.science/pith/SFNMCQOJ6A3HGF55RX73IXS2MO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SFNMCQOJ6A3HGF55RX73IXS2MO/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":"2026-06-10T01:33:47Z","links":{"resolver":"https://pith.science/pith/SFNMCQOJ6A3HGF55RX73IXS2MO","bundle":"https://pith.science/pith/SFNMCQOJ6A3HGF55RX73IXS2MO/bundle.json","state":"https://pith.science/pith/SFNMCQOJ6A3HGF55RX73IXS2MO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SFNMCQOJ6A3HGF55RX73IXS2MO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SFNMCQOJ6A3HGF55RX73IXS2MO","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":"9474ba7237701445deb06e6732cce125cea1d312665e3817a1e8f60242f7cee7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-08-07T12:53:52Z","title_canon_sha256":"994a470dac3d3bb5f9e1d9bbc781cc4ec6d463f6d977f63c21bad67ad5b594c8"},"schema_version":"1.0","source":{"id":"1708.02653","kind":"arxiv","version":22}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.02653","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"1708.02653v22","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.02653","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"SFNMCQOJ6A3H","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"pith_short_16","alias_value":"SFNMCQOJ6A3HGF55","created_at":"2026-06-02T03:04:29Z"},{"alias_kind":"pith_short_8","alias_value":"SFNMCQOJ","created_at":"2026-06-02T03:04:29Z"}],"graph_snapshots":[{"event_id":"sha256:6797df73c9632ca34c3d86770dea21c64a171fbc0ab79323bf18fe378875d640","target":"graph","created_at":"2026-06-02T03:04:29Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1708.02653/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A Hadamard factorisation of the Riemann $\\xi$-function is constructed to characterise the zeros of the Riemann zeta function. The argument proceeds via a probabilistic reformulation: the Riemann Hypothesis is shown to be equivalent to Thorin's condition, namely that $\\xi(\\half)/\\xi(\\half+\\sqrt s\\,)$ is the Laplace transform of a generalised gamma convolution (GGC). A GGC random variable realising this Laplace transform is constructed unconditionally, using L\\'evy representations of the Gamma function, the Euler product for the zeta function, and Ramanujan's Master Theorem, thereby establishing","authors_text":"Nicholas G. Polson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-08-07T12:53:52Z","title":"On Hilbert's 8th Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02653","kind":"arxiv","version":22},"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:f293cee47e1f9777916c0c6f094cc03d903a273de517a26c75f5fc75c103e28a","target":"record","created_at":"2026-06-02T03:04:29Z","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":"9474ba7237701445deb06e6732cce125cea1d312665e3817a1e8f60242f7cee7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-08-07T12:53:52Z","title_canon_sha256":"994a470dac3d3bb5f9e1d9bbc781cc4ec6d463f6d977f63c21bad67ad5b594c8"},"schema_version":"1.0","source":{"id":"1708.02653","kind":"arxiv","version":22}},"canonical_sha256":"915ac141c9f0367317bd8dffb45e5a63a03160f43805176415495511df2e7b91","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"915ac141c9f0367317bd8dffb45e5a63a03160f43805176415495511df2e7b91","first_computed_at":"2026-06-02T03:04:29.524830Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T03:04:29.524830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EnutID6ny0j8bromf9+yRepNzxST374QQ7UNyjx7lXOc7uQcCgPjXqNQ/fIoAjv+9Im+xZ6P9ykg1q30V0tWBw==","signature_status":"signed_v1","signed_at":"2026-06-02T03:04:29.525449Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.02653","source_kind":"arxiv","source_version":22}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f293cee47e1f9777916c0c6f094cc03d903a273de517a26c75f5fc75c103e28a","sha256:6797df73c9632ca34c3d86770dea21c64a171fbc0ab79323bf18fe378875d640"],"state_sha256":"9c1633f3d71f460a1f4bddffb46558d7931bb06c1ef1864da1f606a61a6c1b4e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y1KDTfNoRbYUfeBDedZV9E9EsEkjfvimWJBx+o2PNSUCm0d6MOreoe6aikNRl/6fErUVR38y+dEcHF/95cMxCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T01:33:47.355573Z","bundle_sha256":"f07c0f0a988e6339f7e49bfb2b1d908d3aa960d757504f5566ebb4450064bc4d"}}