{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OW26QSGPKFG7H5NRRF6G6FYGSN","short_pith_number":"pith:OW26QSGP","canonical_record":{"source":{"id":"1607.05517","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T11:03:23Z","cross_cats_sorted":[],"title_canon_sha256":"0d308e9007635291839f3c3f9d6b92ec8b21133612b4e958bc7590dc978ceef4","abstract_canon_sha256":"4c29cc412380a6fc5afbb1bb657db65d07543d4e4971a2859c40775b4513ee49"},"schema_version":"1.0"},"canonical_sha256":"75b5e848cf514df3f5b1897c6f1706934e4eb2287d1df037a46031829b99ea39","source":{"kind":"arxiv","id":"1607.05517","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.05517","created_at":"2026-05-18T00:48:09Z"},{"alias_kind":"arxiv_version","alias_value":"1607.05517v4","created_at":"2026-05-18T00:48:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.05517","created_at":"2026-05-18T00:48:09Z"},{"alias_kind":"pith_short_12","alias_value":"OW26QSGPKFG7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OW26QSGPKFG7H5NR","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OW26QSGP","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OW26QSGPKFG7H5NRRF6G6FYGSN","target":"record","payload":{"canonical_record":{"source":{"id":"1607.05517","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T11:03:23Z","cross_cats_sorted":[],"title_canon_sha256":"0d308e9007635291839f3c3f9d6b92ec8b21133612b4e958bc7590dc978ceef4","abstract_canon_sha256":"4c29cc412380a6fc5afbb1bb657db65d07543d4e4971a2859c40775b4513ee49"},"schema_version":"1.0"},"canonical_sha256":"75b5e848cf514df3f5b1897c6f1706934e4eb2287d1df037a46031829b99ea39","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:09.716461Z","signature_b64":"eQENQQOmrm2c3qSRdyRhuPCo2O7htoegX52Oww6HiFzg5Yn/668KI1xzuMZhMdUn17MT8YHtFji/ksEWeSf1Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"75b5e848cf514df3f5b1897c6f1706934e4eb2287d1df037a46031829b99ea39","last_reissued_at":"2026-05-18T00:48:09.715824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:09.715824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.05517","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:48:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jAl+cs12nyHXkMVpGSbmAk+DM9LA18vUbyBIhH222OuwKBmVwNTlTWIxf5OnOpXxx5a+YJ0oxD4lUk2T9rawAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T05:18:39.630791Z"},"content_sha256":"424e5ce158d97869b5cd26b1d01af798274b35a1c65cc4120c0939fe8cc7f621","schema_version":"1.0","event_id":"sha256:424e5ce158d97869b5cd26b1d01af798274b35a1c65cc4120c0939fe8cc7f621"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OW26QSGPKFG7H5NRRF6G6FYGSN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting primes by sums of frequencies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alejandro Miralles, Dami\\`a Torres","submitted_at":"2016-07-19T11:03:23Z","abstract_excerpt":"We introduce the sequence $(a_n) \\subset (0,1]$ and prove that the asymptotic behaviour of $\\sum_{k=1}^n a_k$ is the same than $\\pi(n)$, the prime-counting function. We also obtain that $\\pi(n) \\sim n a_n$ and we estimate $\\frac{1}{a_n}-\\frac{n}{\\pi(n)}$ showing that $\\lim_{n \\rightarrow \\infty} \\frac{1}{a_n}-\\frac{n}{\\pi(n)}$ is convergent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05517","kind":"arxiv","version":4},"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":"2026-05-18T00:48:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AwS2AZcC/eY9UdiDJ4CR23SuHmNYng8lNS/ORDKOJKNO2n9tdjb1alI24YTZc8VOvJo3tbDc/OtEnUL/DiF/DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T05:18:39.631134Z"},"content_sha256":"1cee69f29a2ddcf81a8bc9715e8f7a00e41fd1611fd50d218dc2bc579d0622a0","schema_version":"1.0","event_id":"sha256:1cee69f29a2ddcf81a8bc9715e8f7a00e41fd1611fd50d218dc2bc579d0622a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OW26QSGPKFG7H5NRRF6G6FYGSN/bundle.json","state_url":"https://pith.science/pith/OW26QSGPKFG7H5NRRF6G6FYGSN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OW26QSGPKFG7H5NRRF6G6FYGSN/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":"2026-06-27T05:18:39Z","links":{"resolver":"https://pith.science/pith/OW26QSGPKFG7H5NRRF6G6FYGSN","bundle":"https://pith.science/pith/OW26QSGPKFG7H5NRRF6G6FYGSN/bundle.json","state":"https://pith.science/pith/OW26QSGPKFG7H5NRRF6G6FYGSN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OW26QSGPKFG7H5NRRF6G6FYGSN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OW26QSGPKFG7H5NRRF6G6FYGSN","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":"4c29cc412380a6fc5afbb1bb657db65d07543d4e4971a2859c40775b4513ee49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T11:03:23Z","title_canon_sha256":"0d308e9007635291839f3c3f9d6b92ec8b21133612b4e958bc7590dc978ceef4"},"schema_version":"1.0","source":{"id":"1607.05517","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.05517","created_at":"2026-05-18T00:48:09Z"},{"alias_kind":"arxiv_version","alias_value":"1607.05517v4","created_at":"2026-05-18T00:48:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.05517","created_at":"2026-05-18T00:48:09Z"},{"alias_kind":"pith_short_12","alias_value":"OW26QSGPKFG7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OW26QSGPKFG7H5NR","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OW26QSGP","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:1cee69f29a2ddcf81a8bc9715e8f7a00e41fd1611fd50d218dc2bc579d0622a0","target":"graph","created_at":"2026-05-18T00:48:09Z","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 introduce the sequence $(a_n) \\subset (0,1]$ and prove that the asymptotic behaviour of $\\sum_{k=1}^n a_k$ is the same than $\\pi(n)$, the prime-counting function. We also obtain that $\\pi(n) \\sim n a_n$ and we estimate $\\frac{1}{a_n}-\\frac{n}{\\pi(n)}$ showing that $\\lim_{n \\rightarrow \\infty} \\frac{1}{a_n}-\\frac{n}{\\pi(n)}$ is convergent.","authors_text":"Alejandro Miralles, Dami\\`a Torres","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T11:03:23Z","title":"Counting primes by sums of frequencies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05517","kind":"arxiv","version":4},"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":"sha256:424e5ce158d97869b5cd26b1d01af798274b35a1c65cc4120c0939fe8cc7f621","target":"record","created_at":"2026-05-18T00:48:09Z","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":"4c29cc412380a6fc5afbb1bb657db65d07543d4e4971a2859c40775b4513ee49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T11:03:23Z","title_canon_sha256":"0d308e9007635291839f3c3f9d6b92ec8b21133612b4e958bc7590dc978ceef4"},"schema_version":"1.0","source":{"id":"1607.05517","kind":"arxiv","version":4}},"canonical_sha256":"75b5e848cf514df3f5b1897c6f1706934e4eb2287d1df037a46031829b99ea39","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"75b5e848cf514df3f5b1897c6f1706934e4eb2287d1df037a46031829b99ea39","first_computed_at":"2026-05-18T00:48:09.715824Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:09.715824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eQENQQOmrm2c3qSRdyRhuPCo2O7htoegX52Oww6HiFzg5Yn/668KI1xzuMZhMdUn17MT8YHtFji/ksEWeSf1Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:09.716461Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.05517","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:424e5ce158d97869b5cd26b1d01af798274b35a1c65cc4120c0939fe8cc7f621","sha256:1cee69f29a2ddcf81a8bc9715e8f7a00e41fd1611fd50d218dc2bc579d0622a0"],"state_sha256":"ad5350eacd476231c5caae744c363c37601939087df1801beb59ca2cef9bf787"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iFugSXuE6wactJY3qH64NbJP0N58gFOkMxizT2Bt2VHcLuVZoh1T7B/8ydK0eMDQGEvRq/rweXpM4iD1G407AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T05:18:39.633068Z","bundle_sha256":"9342fb114f5690affb138767b497ff9e4bdd9ad90cb64afe5a4c1a9c82f898b6"}}