{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:QVNUUNOSPIDBEA3SQSYAUGE7EE","short_pith_number":"pith:QVNUUNOS","schema_version":"1.0","canonical_sha256":"855b4a35d27a0612037284b00a189f21061536878c63578c638bbbc1109b2314","source":{"kind":"arxiv","id":"1509.07188","version":1},"attestation_state":"computed","paper":{"title":"Orderings of weakly correlated random variables, and prime number races with many contestants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Adam J. Harper, Youness Lamzouri","submitted_at":"2015-09-24T00:25:18Z","abstract_excerpt":"We investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI.\n  Among our results we exhibit, for the first time, prime races modulo $q$ with $n$ competitor classes where the biases do not dissolve when $n, q\\to \\infty$. We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of $q$, whereas previous methods could only allow a power of $\\log q$.\n  The proofs use harmonic analysis related to the Hardy--Littlewood circle m"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.07188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-24T00:25:18Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ecaa5b2b2c8497d542f10f85ea2fac755e7d731fcff23201489fca5769f9ea83","abstract_canon_sha256":"69ea27b64c24337bb0606e3917b15a892a706432a57838b8d8a913179f79e06a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:09.708287Z","signature_b64":"xpfIiui/8MAciX+dM1kAKtAkml48G3Hnyb+uXWnUCimilQGw8TuvVe7Hl8PGzHjjsUIeR/dLxU+PqBrHBWf3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"855b4a35d27a0612037284b00a189f21061536878c63578c638bbbc1109b2314","last_reissued_at":"2026-05-18T01:32:09.707390Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:09.707390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orderings of weakly correlated random variables, and prime number races with many contestants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Adam J. Harper, Youness Lamzouri","submitted_at":"2015-09-24T00:25:18Z","abstract_excerpt":"We investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI.\n  Among our results we exhibit, for the first time, prime races modulo $q$ with $n$ competitor classes where the biases do not dissolve when $n, q\\to \\infty$. We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of $q$, whereas previous methods could only allow a power of $\\log q$.\n  The proofs use harmonic analysis related to the Hardy--Littlewood circle m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07188","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.07188","created_at":"2026-05-18T01:32:09.707550+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07188v1","created_at":"2026-05-18T01:32:09.707550+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07188","created_at":"2026-05-18T01:32:09.707550+00:00"},{"alias_kind":"pith_short_12","alias_value":"QVNUUNOSPIDB","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"QVNUUNOSPIDBEA3S","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"QVNUUNOS","created_at":"2026-05-18T12:29:39.896362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE","json":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE.json","graph_json":"https://pith.science/api/pith-number/QVNUUNOSPIDBEA3SQSYAUGE7EE/graph.json","events_json":"https://pith.science/api/pith-number/QVNUUNOSPIDBEA3SQSYAUGE7EE/events.json","paper":"https://pith.science/paper/QVNUUNOS"},"agent_actions":{"view_html":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE","download_json":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE.json","view_paper":"https://pith.science/paper/QVNUUNOS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07188&json=true","fetch_graph":"https://pith.science/api/pith-number/QVNUUNOSPIDBEA3SQSYAUGE7EE/graph.json","fetch_events":"https://pith.science/api/pith-number/QVNUUNOSPIDBEA3SQSYAUGE7EE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE/action/storage_attestation","attest_author":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE/action/author_attestation","sign_citation":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE/action/citation_signature","submit_replication":"https://pith.science/pith/QVNUUNOSPIDBEA3SQSYAUGE7EE/action/replication_record"}},"created_at":"2026-05-18T01:32:09.707550+00:00","updated_at":"2026-05-18T01:32:09.707550+00:00"}