{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:PXO4H3YRYOKCJZWX5S5FQMSTP7","short_pith_number":"pith:PXO4H3YR","schema_version":"1.0","canonical_sha256":"7dddc3ef11c39424e6d7ecba5832537ffb2e840d15002cf93b5e75d497edb178","source":{"kind":"arxiv","id":"2607.00592","version":1},"attestation_state":"computed","paper":{"title":"Character sums over smooth numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu, Seth Hardy","submitted_at":"2026-07-01T08:17:56Z","abstract_excerpt":"Let $\\Psi (x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We show that \\[ \\frac{1}{\\varphi(q)} \\sum_{\\chi \\bmod q} \\Bigl| \\sum_{\\substack{n \\leq x \\\\ P(n) \\leq y}} \\chi(n) \\Bigr| = o \\Bigl( \\sqrt{\\Psi(x,y)} \\Bigr), \\] whenever $(\\log x)^6 \\leq y \\leq x^{\\frac{1}{32 \\log \\log x}}$ and $q \\geq x^{1 + \\varepsilon}$ for some small quantifiable $\\varepsilon > 0$. The saving is substantial when $\\varepsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums."},"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":"2607.00592","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-01T08:17:56Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ce769785b8910cb1284b8b6460e66d24227bed26e99ee4d7be4cf924f5d70aa2","abstract_canon_sha256":"bcf4fd06c2dbb001777960f60eacf116810a4131babb9a112feca769c5cb3cd5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:17:48.458768Z","signature_b64":"2SNFwyQZonL78OOGxlziL4xMkEY359sXNTRT2YT0s8PAERtVk7fRGIZ3r29DSYvi0qjfL8Sn+8LQ2Md28qkgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dddc3ef11c39424e6d7ecba5832537ffb2e840d15002cf93b5e75d497edb178","last_reissued_at":"2026-07-02T01:17:48.458342Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:17:48.458342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Character sums over smooth numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu, Seth Hardy","submitted_at":"2026-07-01T08:17:56Z","abstract_excerpt":"Let $\\Psi (x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We show that \\[ \\frac{1}{\\varphi(q)} \\sum_{\\chi \\bmod q} \\Bigl| \\sum_{\\substack{n \\leq x \\\\ P(n) \\leq y}} \\chi(n) \\Bigr| = o \\Bigl( \\sqrt{\\Psi(x,y)} \\Bigr), \\] whenever $(\\log x)^6 \\leq y \\leq x^{\\frac{1}{32 \\log \\log x}}$ and $q \\geq x^{1 + \\varepsilon}$ for some small quantifiable $\\varepsilon > 0$. The saving is substantial when $\\varepsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00592","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.00592/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.00592","created_at":"2026-07-02T01:17:48.458409+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.00592v1","created_at":"2026-07-02T01:17:48.458409+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00592","created_at":"2026-07-02T01:17:48.458409+00:00"},{"alias_kind":"pith_short_12","alias_value":"PXO4H3YRYOKC","created_at":"2026-07-02T01:17:48.458409+00:00"},{"alias_kind":"pith_short_16","alias_value":"PXO4H3YRYOKCJZWX","created_at":"2026-07-02T01:17:48.458409+00:00"},{"alias_kind":"pith_short_8","alias_value":"PXO4H3YR","created_at":"2026-07-02T01:17:48.458409+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/PXO4H3YRYOKCJZWX5S5FQMSTP7","json":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7.json","graph_json":"https://pith.science/api/pith-number/PXO4H3YRYOKCJZWX5S5FQMSTP7/graph.json","events_json":"https://pith.science/api/pith-number/PXO4H3YRYOKCJZWX5S5FQMSTP7/events.json","paper":"https://pith.science/paper/PXO4H3YR"},"agent_actions":{"view_html":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7","download_json":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7.json","view_paper":"https://pith.science/paper/PXO4H3YR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.00592&json=true","fetch_graph":"https://pith.science/api/pith-number/PXO4H3YRYOKCJZWX5S5FQMSTP7/graph.json","fetch_events":"https://pith.science/api/pith-number/PXO4H3YRYOKCJZWX5S5FQMSTP7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7/action/storage_attestation","attest_author":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7/action/author_attestation","sign_citation":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7/action/citation_signature","submit_replication":"https://pith.science/pith/PXO4H3YRYOKCJZWX5S5FQMSTP7/action/replication_record"}},"created_at":"2026-07-02T01:17:48.458409+00:00","updated_at":"2026-07-02T01:17:48.458409+00:00"}