{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:X5UYGB2A72CWGIVTXBJ7BA4CXN","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":"6a3e5a9112604fed3e3902a4fd723f686aba4696945526ba754321b402939707","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-24T13:54:22Z","title_canon_sha256":"cd05b32b2d2f39c43ae24b441398c23ab591e4ca66b4dda962602dfe95b034c9"},"schema_version":"1.0","source":{"id":"2602.20917","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.20917","created_at":"2026-05-28T02:04:46Z"},{"alias_kind":"arxiv_version","alias_value":"2602.20917v6","created_at":"2026-05-28T02:04:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.20917","created_at":"2026-05-28T02:04:46Z"},{"alias_kind":"pith_short_12","alias_value":"X5UYGB2A72CW","created_at":"2026-05-28T02:04:46Z"},{"alias_kind":"pith_short_16","alias_value":"X5UYGB2A72CWGIVT","created_at":"2026-05-28T02:04:46Z"},{"alias_kind":"pith_short_8","alias_value":"X5UYGB2A","created_at":"2026-05-28T02:04:46Z"}],"graph_snapshots":[{"event_id":"sha256:9f22bc22902f37b1a7d04b431a15b4dead463be6eda07c6ae8aa58104ce105ce","target":"graph","created_at":"2026-05-28T02:04:46Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We obtain some mean value theorems for primes with bilinear forms of moduli up to x^{9/17} or with trilinear forms of moduli up to x^{17/32}. As a by-product, we obtain new upper and lower bounds for π(x; q, a) that hold for almost all moduli q."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The arithmetic information from the cited works of many authors combines with the variants of Harman's sieve to produce majorants and minorants that satisfy the required Bombieri-Vinogradov type mean value theorems without further restrictions."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Refinements of Harman's sieve produce Bombieri-Vinogradov mean value theorems for primes in APs with bilinear moduli up to x^{9/17} and trilinear up to x^{17/32}, yielding new upper and lower bounds for π(x; q, a) for almost all q."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Variants of Harman's sieve produce mean value theorems for primes in arithmetic progressions to moduli as large as x to the 9/17 in bilinear form."}],"snapshot_sha256":"18b592992791f7bd2f5534a355a98912d4995e2908ec2e3006fef083ad40e312"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"afbb8e635ba77297357587632ffc0a791b8a2819e59c42fb6ead33e742a1e859"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2602.20917/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\\mathbb{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\\f","authors_text":"Runbo Li","cross_cats":[],"headline":"Variants of Harman's sieve produce mean value theorems for primes in arithmetic progressions to moduli as large as x to the 9/17 in bilinear form.","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-24T13:54:22Z","title":"Primes in arithmetic progressions to large moduli and refinements of Harman's sieve"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.20917","kind":"arxiv","version":6},"verdict":{"created_at":"2026-05-15T19:56:26.924267Z","id":"b98bb41f-9dd3-4619-9a75-4408348caa72","model_set":{"reader":"grok-4.3"},"one_line_summary":"Refinements of Harman's sieve produce Bombieri-Vinogradov mean value theorems for primes in APs with bilinear moduli up to x^{9/17} and trilinear up to x^{17/32}, yielding new upper and lower bounds for π(x; q, a) for almost all q.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Variants of Harman's sieve produce mean value theorems for primes in arithmetic progressions to moduli as large as x to the 9/17 in bilinear form.","strongest_claim":"We obtain some mean value theorems for primes with bilinear forms of moduli up to x^{9/17} or with trilinear forms of moduli up to x^{17/32}. As a by-product, we obtain new upper and lower bounds for π(x; q, a) that hold for almost all moduli q.","weakest_assumption":"The arithmetic information from the cited works of many authors combines with the variants of Harman's sieve to produce majorants and minorants that satisfy the required Bombieri-Vinogradov type mean value theorems without further restrictions."}},"verdict_id":"b98bb41f-9dd3-4619-9a75-4408348caa72"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2308b11615b8ac3971a5b915a0c8518a08d557dd02b41e2fd30d83dd90f43ffa","target":"record","created_at":"2026-05-28T02:04:46Z","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":"6a3e5a9112604fed3e3902a4fd723f686aba4696945526ba754321b402939707","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-24T13:54:22Z","title_canon_sha256":"cd05b32b2d2f39c43ae24b441398c23ab591e4ca66b4dda962602dfe95b034c9"},"schema_version":"1.0","source":{"id":"2602.20917","kind":"arxiv","version":6}},"canonical_sha256":"bf69830740fe856322b3b853f08382bb7646b56088c7d88026395268fadc6484","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf69830740fe856322b3b853f08382bb7646b56088c7d88026395268fadc6484","first_computed_at":"2026-05-28T02:04:46.485827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T02:04:46.485827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u4vKtqS8xHd4Q2GCGRWrv8SBER3m/zdkQdMbIhhj+3SgCrdjLM7CqiFrKN8DcgVcXPHMHYFiZQkc/aHx7AZmAQ==","signature_status":"signed_v1","signed_at":"2026-05-28T02:04:46.486335Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.20917","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2308b11615b8ac3971a5b915a0c8518a08d557dd02b41e2fd30d83dd90f43ffa","sha256:9f22bc22902f37b1a7d04b431a15b4dead463be6eda07c6ae8aa58104ce105ce"],"state_sha256":"370c723f9d6aefbeed842f77f8134a5a1e2ff5da8c971f5ac9e3549719befdb9"}