{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:2ZYOS6OTTFAIJ53I2ZFJ62B3SQ","short_pith_number":"pith:2ZYOS6OT","canonical_record":{"source":{"id":"1206.1613","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-07T20:56:27Z","cross_cats_sorted":[],"title_canon_sha256":"96cb1811f8722415d62a34d43855c9c52c0a661c732fa9e5adf5142bac5362d3","abstract_canon_sha256":"6b2b9561f1ca15ddb7b5399746b0cd0861889d818c6ebb962565db4919b73a75"},"schema_version":"1.0"},"canonical_sha256":"d670e979d3994084f768d64a9f683b94178219831d4d3742107daa81c91fdf32","source":{"kind":"arxiv","id":"1206.1613","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1613","created_at":"2026-05-18T03:54:03Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1613v1","created_at":"2026-05-18T03:54:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1613","created_at":"2026-05-18T03:54:03Z"},{"alias_kind":"pith_short_12","alias_value":"2ZYOS6OTTFAI","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2ZYOS6OTTFAIJ53I","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2ZYOS6OT","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:2ZYOS6OTTFAIJ53I2ZFJ62B3SQ","target":"record","payload":{"canonical_record":{"source":{"id":"1206.1613","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-07T20:56:27Z","cross_cats_sorted":[],"title_canon_sha256":"96cb1811f8722415d62a34d43855c9c52c0a661c732fa9e5adf5142bac5362d3","abstract_canon_sha256":"6b2b9561f1ca15ddb7b5399746b0cd0861889d818c6ebb962565db4919b73a75"},"schema_version":"1.0"},"canonical_sha256":"d670e979d3994084f768d64a9f683b94178219831d4d3742107daa81c91fdf32","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:03.710370Z","signature_b64":"aU96Pv6Fxqn2WLH2qevQFHlTwCRe9E+kV7Z9KdUGfe9/Oh6w1oC2jXJdkcr1yTpUkwQzE35cI1al7PlBtfoQDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d670e979d3994084f768d64a9f683b94178219831d4d3742107daa81c91fdf32","last_reissued_at":"2026-05-18T03:54:03.709800Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:03.709800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.1613","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":"2026-05-18T03:54:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZZvfDKlVAhO+J2qvIMSm1zLGmWgOpqByKA7BGMvP9zJEf/nLrxLgOf4aeOCBsjeM7GYlaKLh0/HwK6p95Ws5Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:12:37.817621Z"},"content_sha256":"6fb09dd63bd4fe9575604edad1b4767436c9eb2505584be70c41425fb78ab53a","schema_version":"1.0","event_id":"sha256:6fb09dd63bd4fe9575604edad1b4767436c9eb2505584be70c41425fb78ab53a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:2ZYOS6OTTFAIJ53I2ZFJ62B3SQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Average Number of Lattice Points in a Disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Robert S. Strichartz, Sujay Jayakar","submitted_at":"2012-06-07T20:56:27Z","abstract_excerpt":"The difference between the number of lattice points in a disk of radius $\\sqrt{t}/2\\pi$ and the area of the disk $t/4\\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \\leq t \\leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1613","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":"2026-05-18T03:54:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QJS7f9tHjws4M5XUtwV55QveVutsNChrIE0uMbChge2iEL/EsLi/Oe3/1qpIOQM2LmcCgoZz9AtMk30Ps1FADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:12:37.817965Z"},"content_sha256":"1db8ce80196dd463b782ce934287fb8af6d1e219dc3af1748069d9d8c4603b84","schema_version":"1.0","event_id":"sha256:1db8ce80196dd463b782ce934287fb8af6d1e219dc3af1748069d9d8c4603b84"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ/bundle.json","state_url":"https://pith.science/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ/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":"2026-06-02T23:12:37Z","links":{"resolver":"https://pith.science/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ","bundle":"https://pith.science/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ/bundle.json","state":"https://pith.science/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2ZYOS6OTTFAIJ53I2ZFJ62B3SQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:2ZYOS6OTTFAIJ53I2ZFJ62B3SQ","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":"6b2b9561f1ca15ddb7b5399746b0cd0861889d818c6ebb962565db4919b73a75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-07T20:56:27Z","title_canon_sha256":"96cb1811f8722415d62a34d43855c9c52c0a661c732fa9e5adf5142bac5362d3"},"schema_version":"1.0","source":{"id":"1206.1613","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1613","created_at":"2026-05-18T03:54:03Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1613v1","created_at":"2026-05-18T03:54:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1613","created_at":"2026-05-18T03:54:03Z"},{"alias_kind":"pith_short_12","alias_value":"2ZYOS6OTTFAI","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2ZYOS6OTTFAIJ53I","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2ZYOS6OT","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:1db8ce80196dd463b782ce934287fb8af6d1e219dc3af1748069d9d8c4603b84","target":"graph","created_at":"2026-05-18T03:54:03Z","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":"The difference between the number of lattice points in a disk of radius $\\sqrt{t}/2\\pi$ and the area of the disk $t/4\\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \\leq t \\leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.","authors_text":"Robert S. Strichartz, Sujay Jayakar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-07T20:56:27Z","title":"Average Number of Lattice Points in a Disk"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1613","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":"sha256:6fb09dd63bd4fe9575604edad1b4767436c9eb2505584be70c41425fb78ab53a","target":"record","created_at":"2026-05-18T03:54:03Z","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":"6b2b9561f1ca15ddb7b5399746b0cd0861889d818c6ebb962565db4919b73a75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-07T20:56:27Z","title_canon_sha256":"96cb1811f8722415d62a34d43855c9c52c0a661c732fa9e5adf5142bac5362d3"},"schema_version":"1.0","source":{"id":"1206.1613","kind":"arxiv","version":1}},"canonical_sha256":"d670e979d3994084f768d64a9f683b94178219831d4d3742107daa81c91fdf32","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d670e979d3994084f768d64a9f683b94178219831d4d3742107daa81c91fdf32","first_computed_at":"2026-05-18T03:54:03.709800Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:03.709800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aU96Pv6Fxqn2WLH2qevQFHlTwCRe9E+kV7Z9KdUGfe9/Oh6w1oC2jXJdkcr1yTpUkwQzE35cI1al7PlBtfoQDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:03.710370Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.1613","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6fb09dd63bd4fe9575604edad1b4767436c9eb2505584be70c41425fb78ab53a","sha256:1db8ce80196dd463b782ce934287fb8af6d1e219dc3af1748069d9d8c4603b84"],"state_sha256":"225d44a972ceb130b3bd01a9759fe67df394d3ff4a9ac7415db547747abafc66"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B4sJKUgXotVWz/YbnZvQlKkzIpAe9/LrPm/YiXmUA76ipX15Bv49nICqYvJH9nfYDG0Ov1Mk0ti7+C3oRCpTBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T23:12:37.819926Z","bundle_sha256":"7218577eca44b65287281a23210441200e74814f080530315698a16c02c34ac7"}}