{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:SGTBCBNF7L4RLB5UYQLCE2IGTG","short_pith_number":"pith:SGTBCBNF","canonical_record":{"source":{"id":"2411.18903","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-28T04:27:09Z","cross_cats_sorted":[],"title_canon_sha256":"b41fac976b831b9280e81d528502e61bc6b3c4871d9a0b0f5ff4d4a74d05c045","abstract_canon_sha256":"7940a6fa8663162f7d0b256ad3dbe9d2e1ca6339d5f493c0384dd55fe7a4cc4a"},"schema_version":"1.0"},"canonical_sha256":"91a61105a5faf91587b4c41622690699b721ff009838797af6fd91437b5d762a","source":{"kind":"arxiv","id":"2411.18903","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.18903","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"arxiv_version","alias_value":"2411.18903v2","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.18903","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"pith_short_12","alias_value":"SGTBCBNF7L4R","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"SGTBCBNF7L4RLB5U","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"SGTBCBNF","created_at":"2026-07-05T11:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:SGTBCBNF7L4RLB5UYQLCE2IGTG","target":"record","payload":{"canonical_record":{"source":{"id":"2411.18903","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-28T04:27:09Z","cross_cats_sorted":[],"title_canon_sha256":"b41fac976b831b9280e81d528502e61bc6b3c4871d9a0b0f5ff4d4a74d05c045","abstract_canon_sha256":"7940a6fa8663162f7d0b256ad3dbe9d2e1ca6339d5f493c0384dd55fe7a4cc4a"},"schema_version":"1.0"},"canonical_sha256":"91a61105a5faf91587b4c41622690699b721ff009838797af6fd91437b5d762a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:26:24.623135Z","signature_b64":"g2THRPySn6jw8gVAD1rIPwAa+fF3Ir9gGsN/knAykmYt/+1U1SMTv48CcsEtGWk8nJUx14PMrw+gOBNj6yDyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91a61105a5faf91587b4c41622690699b721ff009838797af6fd91437b5d762a","last_reissued_at":"2026-07-05T11:26:24.622596Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:26:24.622596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.18903","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":"2026-07-05T11:26:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dLpALT1+PMhXIWYRLD84p5tRiAsg5Si4IRZAuCmsGs4ITgrpMD01+CUm9c5PocRRvr6gxGOiW+CM1C44GVYTAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:59:56.363806Z"},"content_sha256":"0252bfaef5810f162ff010f241b08b2eb4e280e4954468b3a47d341553b9e78d","schema_version":"1.0","event_id":"sha256:0252bfaef5810f162ff010f241b08b2eb4e280e4954468b3a47d341553b9e78d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:SGTBCBNF7L4RLB5UYQLCE2IGTG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the mean values of the error terms in Mertens' theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tianyu Zhao","submitted_at":"2024-11-28T04:27:09Z","abstract_excerpt":"For $i\\in \\{1,2,3\\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i\\in \\{1,2\\}$ the Riemann hypothesis is equivalent to the condition $\\int_2^X E_i(x) \\:\\mathrm{d}x>0$ for all $X>2$, and we examine assumptions under which the equivalence also holds for $i=3$. In addition, we extend our results to analogues of Mertens' theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.18903","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.18903/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":"2026-07-05T11:26:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6klimTnc7egDWywVvKhD+E9WJ/FZx3uRQLHdspCEbMREh3ZiH4FWi7rA6plHDVY00jG6gJuN+ngbmqvssZveBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:59:56.364756Z"},"content_sha256":"e2abf6701aeae30bfc0927f798c17e537d4e42943301d96efc7795ef659a5618","schema_version":"1.0","event_id":"sha256:e2abf6701aeae30bfc0927f798c17e537d4e42943301d96efc7795ef659a5618"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG/bundle.json","state_url":"https://pith.science/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG/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":"2026-08-12T18:59:56Z","links":{"resolver":"https://pith.science/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG","bundle":"https://pith.science/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG/bundle.json","state":"https://pith.science/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SGTBCBNF7L4RLB5UYQLCE2IGTG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:SGTBCBNF7L4RLB5UYQLCE2IGTG","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":"7940a6fa8663162f7d0b256ad3dbe9d2e1ca6339d5f493c0384dd55fe7a4cc4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-28T04:27:09Z","title_canon_sha256":"b41fac976b831b9280e81d528502e61bc6b3c4871d9a0b0f5ff4d4a74d05c045"},"schema_version":"1.0","source":{"id":"2411.18903","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.18903","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"arxiv_version","alias_value":"2411.18903v2","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.18903","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"pith_short_12","alias_value":"SGTBCBNF7L4R","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"SGTBCBNF7L4RLB5U","created_at":"2026-07-05T11:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"SGTBCBNF","created_at":"2026-07-05T11:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:e2abf6701aeae30bfc0927f798c17e537d4e42943301d96efc7795ef659a5618","target":"graph","created_at":"2026-07-05T11:26:24Z","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/2411.18903/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For $i\\in \\{1,2,3\\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i\\in \\{1,2\\}$ the Riemann hypothesis is equivalent to the condition $\\int_2^X E_i(x) \\:\\mathrm{d}x>0$ for all $X>2$, and we examine assumptions under which the equivalence also holds for $i=3$. In addition, we extend our results to analogues of Mertens' theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.","authors_text":"Tianyu Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-28T04:27:09Z","title":"On the mean values of the error terms in Mertens' theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.18903","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":"sha256:0252bfaef5810f162ff010f241b08b2eb4e280e4954468b3a47d341553b9e78d","target":"record","created_at":"2026-07-05T11:26:24Z","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":"7940a6fa8663162f7d0b256ad3dbe9d2e1ca6339d5f493c0384dd55fe7a4cc4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-28T04:27:09Z","title_canon_sha256":"b41fac976b831b9280e81d528502e61bc6b3c4871d9a0b0f5ff4d4a74d05c045"},"schema_version":"1.0","source":{"id":"2411.18903","kind":"arxiv","version":2}},"canonical_sha256":"91a61105a5faf91587b4c41622690699b721ff009838797af6fd91437b5d762a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"91a61105a5faf91587b4c41622690699b721ff009838797af6fd91437b5d762a","first_computed_at":"2026-07-05T11:26:24.622596Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:26:24.622596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g2THRPySn6jw8gVAD1rIPwAa+fF3Ir9gGsN/knAykmYt/+1U1SMTv48CcsEtGWk8nJUx14PMrw+gOBNj6yDyBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:26:24.623135Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.18903","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0252bfaef5810f162ff010f241b08b2eb4e280e4954468b3a47d341553b9e78d","sha256:e2abf6701aeae30bfc0927f798c17e537d4e42943301d96efc7795ef659a5618"],"state_sha256":"7ea7aaf79dfd959cefb31af686dd863bf0006a7ed886a634fd28a34109284405"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mm953wfjMO4ZbulDokVEVW8cfULRY+BX0x0Q3DkGa+icfG3L4WQ7YhRD+9ooFsVJeMUEYwrd4jvIXa/sJNYHAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T18:59:56.372791Z","bundle_sha256":"d18dd9630443a0b457a4aa49d0d4a5057f65cb23703c365fbcdc6cff1141973b"}}