{"paper":{"title":"Primes in arithmetic progressions to large moduli and refinements of Harman's sieve","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","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.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Runbo Li","submitted_at":"2026-02-24T13:54:22Z","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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"18b592992791f7bd2f5534a355a98912d4995e2908ec2e3006fef083ad40e312"},"source":{"id":"2602.20917","kind":"arxiv","version":6},"verdict":{"id":"b98bb41f-9dd3-4619-9a75-4408348caa72","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T19:56:26.924267Z","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.","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","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.","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.20917/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":2,"snapshot_sha256":"afbb8e635ba77297357587632ffc0a791b8a2819e59c42fb6ead33e742a1e859"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}