{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7A6TWUWXYM47CO53JZJAUSRGXM","short_pith_number":"pith:7A6TWUWX","canonical_record":{"source":{"id":"1601.05324","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-20T16:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"36ee819cf87d8ba239627801106702ae77cd1c96653f7a87f4fd2e8f6f01207f","abstract_canon_sha256":"9e24dee6cd0c121a7a683b10d197f3c127449f8c65347c0e1a05694cd9f74135"},"schema_version":"1.0"},"canonical_sha256":"f83d3b52d7c339f13bbb4e520a4a26bb243a976f4e74444ab8b8aa501e1f96c7","source":{"kind":"arxiv","id":"1601.05324","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05324","created_at":"2026-05-18T00:36:51Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05324v2","created_at":"2026-05-18T00:36:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05324","created_at":"2026-05-18T00:36:51Z"},{"alias_kind":"pith_short_12","alias_value":"7A6TWUWXYM47","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7A6TWUWXYM47CO53","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7A6TWUWX","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7A6TWUWXYM47CO53JZJAUSRGXM","target":"record","payload":{"canonical_record":{"source":{"id":"1601.05324","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-20T16:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"36ee819cf87d8ba239627801106702ae77cd1c96653f7a87f4fd2e8f6f01207f","abstract_canon_sha256":"9e24dee6cd0c121a7a683b10d197f3c127449f8c65347c0e1a05694cd9f74135"},"schema_version":"1.0"},"canonical_sha256":"f83d3b52d7c339f13bbb4e520a4a26bb243a976f4e74444ab8b8aa501e1f96c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:51.720533Z","signature_b64":"Ck9Te+PvJD6FX9R603CktkGmNSMFQT0G1sshxZRf518bFUTMitnJZqFuteIh1EDmJW8n14t/3stnLPJZifd2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f83d3b52d7c339f13bbb4e520a4a26bb243a976f4e74444ab8b8aa501e1f96c7","last_reissued_at":"2026-05-18T00:36:51.719843Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:51.719843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.05324","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:36:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YJXqWBnzjx0jf0AzPsXdzP49QBw+/1+TfcftUd4HvdVCVrCIenizXrutxmgxTiwPGL3VAVEu7qqq22xTv+GYAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T01:31:58.084817Z"},"content_sha256":"0b8f064985abe7baec1c9193b9e77db8dead63c4dff31489985493b5eaecfe32","schema_version":"1.0","event_id":"sha256:0b8f064985abe7baec1c9193b9e77db8dead63c4dff31489985493b5eaecfe32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7A6TWUWXYM47CO53JZJAUSRGXM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On General Prime Number Theorems with Remainder","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gregory Debruyne, Jasson Vindas","submitted_at":"2016-01-20T16:42:58Z","abstract_excerpt":"We show that for Beurling generalized numbers the prime number theorem in remainder form $$\\pi(x) = \\operatorname*{Li}(x) + O\\left(\\frac{x}{\\log^{n}x}\\right) \\quad \\mbox{for all } n\\in\\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\\left(\\frac{x}{\\log^{n}x}\\right) \\quad \\mbox{for all } n \\in \\mathbb{N},$$ where $N$ and $\\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime numbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05324","kind":"arxiv","version":2},"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":"2026-05-18T00:36:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ktB0E3zqek58yMbH2pFBqxvHtlQfBij3874PheGFlGfMK0YMVA/3+1ADSefD4kkOLpGxe+P39ILjRVu3B/mfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T01:31:58.085446Z"},"content_sha256":"5adc924aea36ad8a1509ddb274dc2cfa8aeb3c5734633aa3d3f73a7201915278","schema_version":"1.0","event_id":"sha256:5adc924aea36ad8a1509ddb274dc2cfa8aeb3c5734633aa3d3f73a7201915278"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7A6TWUWXYM47CO53JZJAUSRGXM/bundle.json","state_url":"https://pith.science/pith/7A6TWUWXYM47CO53JZJAUSRGXM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7A6TWUWXYM47CO53JZJAUSRGXM/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":"2026-05-28T01:31:58Z","links":{"resolver":"https://pith.science/pith/7A6TWUWXYM47CO53JZJAUSRGXM","bundle":"https://pith.science/pith/7A6TWUWXYM47CO53JZJAUSRGXM/bundle.json","state":"https://pith.science/pith/7A6TWUWXYM47CO53JZJAUSRGXM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7A6TWUWXYM47CO53JZJAUSRGXM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7A6TWUWXYM47CO53JZJAUSRGXM","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":"9e24dee6cd0c121a7a683b10d197f3c127449f8c65347c0e1a05694cd9f74135","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-20T16:42:58Z","title_canon_sha256":"36ee819cf87d8ba239627801106702ae77cd1c96653f7a87f4fd2e8f6f01207f"},"schema_version":"1.0","source":{"id":"1601.05324","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05324","created_at":"2026-05-18T00:36:51Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05324v2","created_at":"2026-05-18T00:36:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05324","created_at":"2026-05-18T00:36:51Z"},{"alias_kind":"pith_short_12","alias_value":"7A6TWUWXYM47","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7A6TWUWXYM47CO53","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7A6TWUWX","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:5adc924aea36ad8a1509ddb274dc2cfa8aeb3c5734633aa3d3f73a7201915278","target":"graph","created_at":"2026-05-18T00:36:51Z","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":"We show that for Beurling generalized numbers the prime number theorem in remainder form $$\\pi(x) = \\operatorname*{Li}(x) + O\\left(\\frac{x}{\\log^{n}x}\\right) \\quad \\mbox{for all } n\\in\\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\\left(\\frac{x}{\\log^{n}x}\\right) \\quad \\mbox{for all } n \\in \\mathbb{N},$$ where $N$ and $\\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime numbe","authors_text":"Gregory Debruyne, Jasson Vindas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-20T16:42:58Z","title":"On General Prime Number Theorems with Remainder"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05324","kind":"arxiv","version":2},"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":"sha256:0b8f064985abe7baec1c9193b9e77db8dead63c4dff31489985493b5eaecfe32","target":"record","created_at":"2026-05-18T00:36:51Z","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":"9e24dee6cd0c121a7a683b10d197f3c127449f8c65347c0e1a05694cd9f74135","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-20T16:42:58Z","title_canon_sha256":"36ee819cf87d8ba239627801106702ae77cd1c96653f7a87f4fd2e8f6f01207f"},"schema_version":"1.0","source":{"id":"1601.05324","kind":"arxiv","version":2}},"canonical_sha256":"f83d3b52d7c339f13bbb4e520a4a26bb243a976f4e74444ab8b8aa501e1f96c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f83d3b52d7c339f13bbb4e520a4a26bb243a976f4e74444ab8b8aa501e1f96c7","first_computed_at":"2026-05-18T00:36:51.719843Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:36:51.719843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ck9Te+PvJD6FX9R603CktkGmNSMFQT0G1sshxZRf518bFUTMitnJZqFuteIh1EDmJW8n14t/3stnLPJZifd2Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:36:51.720533Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.05324","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b8f064985abe7baec1c9193b9e77db8dead63c4dff31489985493b5eaecfe32","sha256:5adc924aea36ad8a1509ddb274dc2cfa8aeb3c5734633aa3d3f73a7201915278"],"state_sha256":"f080e0cc8ccba429d3bcb4ef7a1baecd1526a5252d574d8fb5939a459b9cd85f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4knh/rG8eav9bopcOiXrgE35tfe5tarV82mS0W/YqB9rjJRZOP4NwM+yEC5U1JbpUytcn7YkPcM5q5+Sf8mjDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T01:31:58.088247Z","bundle_sha256":"fdade9412b93a59bb5163c77f7d48d93f7067124dbe5c3eb324b609f1c78f903"}}