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In this paper, we shall prove that $$ \\sum_{n\\leq N}f(n)\\chi(n+a)\\ll {N\\over q^{1\\over 4}}\\log\\log(6N)+q^{1\\over 4}N^{1\\over 2}\\log(6N)+{N\\over \\sqrt{\\log\\log(6N)}}. $$\n  We shall also prove that \\begin{align*} &\\sum_{n\\leq N}f(n)\\chi(n+a_1)\\cdots\\chi(n+a_t)\\ll {N\\over q^{1\\over 4}}\\log\\log(6N)\\\\ &\\quad+q^{1\\over 4}N^{1\\over 2}\\log(6N)+{N\\over \\sqrt{\\log\\log(6N)}}, \\end{align*} where $t\\geq 2$, $a"},"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":"1404.2204","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-08T16:45:40Z","cross_cats_sorted":[],"title_canon_sha256":"16c85335b674d5026c1ecddcd6f51d88ea37b725e4e61501594affdf6bab1781","abstract_canon_sha256":"55ccadad43f3e514b1e7093ba6d97760eb8e89034dbcb11c64737f225cf010e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:25.775773Z","signature_b64":"HT/T9inEApnNos1Qwc4Avwrc+Knodyz5PkjewEi0tFC5Kt0ua8OUhSAsewkyHe4VoXK6FqC+Ym91Zqra2VTCCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"79eddfb62efa25ab51d47bce4a2e1c611c6c9736d6b0bfd59b84f2cf2e37ec94","last_reissued_at":"2026-05-18T02:48:25.775106Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:25.775106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shifted Character Sums with Multiplicative Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chaohua Jia, Ke Gong","submitted_at":"2014-04-08T16:45:40Z","abstract_excerpt":"Let $f(n)$ be a multiplicative function satisfying $|f(n)|\\leq 1$, $q$ $(\\leq N^2)$ be a prime number and $a$ be an integer with $(a,\\,q)=1$, $\\chi$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$ \\sum_{n\\leq N}f(n)\\chi(n+a)\\ll {N\\over q^{1\\over 4}}\\log\\log(6N)+q^{1\\over 4}N^{1\\over 2}\\log(6N)+{N\\over \\sqrt{\\log\\log(6N)}}. $$\n  We shall also prove that \\begin{align*} &\\sum_{n\\leq N}f(n)\\chi(n+a_1)\\cdots\\chi(n+a_t)\\ll {N\\over q^{1\\over 4}}\\log\\log(6N)\\\\ &\\quad+q^{1\\over 4}N^{1\\over 2}\\log(6N)+{N\\over \\sqrt{\\log\\log(6N)}}, \\end{align*} where $t\\geq 2$, $a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2204","kind":"arxiv","version":2},"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":"1404.2204","created_at":"2026-05-18T02:48:25.775200+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.2204v2","created_at":"2026-05-18T02:48:25.775200+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.2204","created_at":"2026-05-18T02:48:25.775200+00:00"},{"alias_kind":"pith_short_12","alias_value":"PHW57NRO7IS2","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"PHW57NRO7IS2WUOU","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"PHW57NRO","created_at":"2026-05-18T12:28:43.426989+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/PHW57NRO7IS2WUOUPPHEULQ4ME","json":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME.json","graph_json":"https://pith.science/api/pith-number/PHW57NRO7IS2WUOUPPHEULQ4ME/graph.json","events_json":"https://pith.science/api/pith-number/PHW57NRO7IS2WUOUPPHEULQ4ME/events.json","paper":"https://pith.science/paper/PHW57NRO"},"agent_actions":{"view_html":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME","download_json":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME.json","view_paper":"https://pith.science/paper/PHW57NRO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.2204&json=true","fetch_graph":"https://pith.science/api/pith-number/PHW57NRO7IS2WUOUPPHEULQ4ME/graph.json","fetch_events":"https://pith.science/api/pith-number/PHW57NRO7IS2WUOUPPHEULQ4ME/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME/action/storage_attestation","attest_author":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME/action/author_attestation","sign_citation":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME/action/citation_signature","submit_replication":"https://pith.science/pith/PHW57NRO7IS2WUOUPPHEULQ4ME/action/replication_record"}},"created_at":"2026-05-18T02:48:25.775200+00:00","updated_at":"2026-05-18T02:48:25.775200+00:00"}