{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:VE7XHENR2AYBSUXO22J4NL3REG","short_pith_number":"pith:VE7XHENR","canonical_record":{"source":{"id":"2305.17597","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-05-28T00:00:56Z","cross_cats_sorted":[],"title_canon_sha256":"041b36bc07073bcfdd529ca909243c9b73638db010dcf92012a9b9dc8b8ce2ac","abstract_canon_sha256":"af9c635cb1a11a2c307b4e78ebba6ebdfaca377f7825dbe814a1be9466890748"},"schema_version":"1.0"},"canonical_sha256":"a93f7391b1d0301952eed693c6af7121932d77541a233936e6bf77bdc4f84ab7","source":{"kind":"arxiv","id":"2305.17597","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2305.17597","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"2305.17597v1","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.17597","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"VE7XHENR2AYB","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"pith_short_16","alias_value":"VE7XHENR2AYBSUXO","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"pith_short_8","alias_value":"VE7XHENR","created_at":"2026-07-05T06:14:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:VE7XHENR2AYBSUXO22J4NL3REG","target":"record","payload":{"canonical_record":{"source":{"id":"2305.17597","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-05-28T00:00:56Z","cross_cats_sorted":[],"title_canon_sha256":"041b36bc07073bcfdd529ca909243c9b73638db010dcf92012a9b9dc8b8ce2ac","abstract_canon_sha256":"af9c635cb1a11a2c307b4e78ebba6ebdfaca377f7825dbe814a1be9466890748"},"schema_version":"1.0"},"canonical_sha256":"a93f7391b1d0301952eed693c6af7121932d77541a233936e6bf77bdc4f84ab7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:14:46.202259Z","signature_b64":"9uJOl5Ysv90PhChmLObdKa8hRC+nCU4xe/hRlHFgprZYDUh2/Mhwi5jWrjS9nD7CW5Dk5ghTvY+dPP2CfeOOCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a93f7391b1d0301952eed693c6af7121932d77541a233936e6bf77bdc4f84ab7","last_reissued_at":"2026-07-05T06:14:46.201933Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:14:46.201933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2305.17597","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":"2026-07-05T06:14:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8CHcZCGUoAlYWcXErPI742XUiYXF4ASqiTDUIpy/Z5RzLh45pDqQlPGElQYUCXGtUGyLU/j2YuO8De8srMCPBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T09:13:12.768789Z"},"content_sha256":"1d71854c88e3262b9c5b2873cd288bd362c403eaddbab4c49bfa59ec7eafa549","schema_version":"1.0","event_id":"sha256:1d71854c88e3262b9c5b2873cd288bd362c403eaddbab4c49bfa59ec7eafa549"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:VE7XHENR2AYBSUXO22J4NL3REG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Ramanujan-Fourier expansions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David T. Nguyen","submitted_at":"2023-05-28T00:00:56Z","abstract_excerpt":"We heuristically study the shifted convolution $\\sum_{n\\le X} \\tau_k(n) \\tau_\\ell(n+h)$ using a normalized version of Ramanujan-Fourier expansions for $\\tau_k(n)$ and verify they produce the expected answer."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.17597","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.17597/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":"2026-07-05T06:14:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z0z8/dcIynY48BbP8E7O2TkQb587L7HHSdwwiLoRyyotp4LfvwzRALEJMm1BcDAMs4lOtLcwREPrBq/0ViehDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T09:13:12.769686Z"},"content_sha256":"5ca7ec15014ccde0dccb661e63ac7dc07d49e4fb7dbdd492b8e9b67a7a0882e3","schema_version":"1.0","event_id":"sha256:5ca7ec15014ccde0dccb661e63ac7dc07d49e4fb7dbdd492b8e9b67a7a0882e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VE7XHENR2AYBSUXO22J4NL3REG/bundle.json","state_url":"https://pith.science/pith/VE7XHENR2AYBSUXO22J4NL3REG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VE7XHENR2AYBSUXO22J4NL3REG/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":"2026-08-11T09:13:12Z","links":{"resolver":"https://pith.science/pith/VE7XHENR2AYBSUXO22J4NL3REG","bundle":"https://pith.science/pith/VE7XHENR2AYBSUXO22J4NL3REG/bundle.json","state":"https://pith.science/pith/VE7XHENR2AYBSUXO22J4NL3REG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VE7XHENR2AYBSUXO22J4NL3REG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:VE7XHENR2AYBSUXO22J4NL3REG","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":"af9c635cb1a11a2c307b4e78ebba6ebdfaca377f7825dbe814a1be9466890748","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-05-28T00:00:56Z","title_canon_sha256":"041b36bc07073bcfdd529ca909243c9b73638db010dcf92012a9b9dc8b8ce2ac"},"schema_version":"1.0","source":{"id":"2305.17597","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2305.17597","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"2305.17597v1","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.17597","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"VE7XHENR2AYB","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"pith_short_16","alias_value":"VE7XHENR2AYBSUXO","created_at":"2026-07-05T06:14:46Z"},{"alias_kind":"pith_short_8","alias_value":"VE7XHENR","created_at":"2026-07-05T06:14:46Z"}],"graph_snapshots":[{"event_id":"sha256:5ca7ec15014ccde0dccb661e63ac7dc07d49e4fb7dbdd492b8e9b67a7a0882e3","target":"graph","created_at":"2026-07-05T06:14:46Z","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/2305.17597/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We heuristically study the shifted convolution $\\sum_{n\\le X} \\tau_k(n) \\tau_\\ell(n+h)$ using a normalized version of Ramanujan-Fourier expansions for $\\tau_k(n)$ and verify they produce the expected answer.","authors_text":"David T. Nguyen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-05-28T00:00:56Z","title":"On Ramanujan-Fourier expansions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.17597","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":"sha256:1d71854c88e3262b9c5b2873cd288bd362c403eaddbab4c49bfa59ec7eafa549","target":"record","created_at":"2026-07-05T06:14:46Z","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":"af9c635cb1a11a2c307b4e78ebba6ebdfaca377f7825dbe814a1be9466890748","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-05-28T00:00:56Z","title_canon_sha256":"041b36bc07073bcfdd529ca909243c9b73638db010dcf92012a9b9dc8b8ce2ac"},"schema_version":"1.0","source":{"id":"2305.17597","kind":"arxiv","version":1}},"canonical_sha256":"a93f7391b1d0301952eed693c6af7121932d77541a233936e6bf77bdc4f84ab7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a93f7391b1d0301952eed693c6af7121932d77541a233936e6bf77bdc4f84ab7","first_computed_at":"2026-07-05T06:14:46.201933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:14:46.201933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9uJOl5Ysv90PhChmLObdKa8hRC+nCU4xe/hRlHFgprZYDUh2/Mhwi5jWrjS9nD7CW5Dk5ghTvY+dPP2CfeOOCA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:14:46.202259Z","signed_message":"canonical_sha256_bytes"},"source_id":"2305.17597","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d71854c88e3262b9c5b2873cd288bd362c403eaddbab4c49bfa59ec7eafa549","sha256:5ca7ec15014ccde0dccb661e63ac7dc07d49e4fb7dbdd492b8e9b67a7a0882e3"],"state_sha256":"58b36412b9756f4cbe65c3e3115fc5ec483a21a90759f47e10aa7c14dcd88fa4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HWXb4ukRynMXgUvlOYOPDI8dSpr7/KtLXc+qu2GUMt+jwGJVpnFQvJE0+MuHZVAYN/KkAUd5B8aM9xDrnA8oCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T09:13:12.775979Z","bundle_sha256":"f529a062cac737b81cde0449b73cdce70a829b1cf64bae626da94cf10d0d6718"}}