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In this paper, we shall prove that $$\\sum_{\\substack{n\\leq N\\\\ (n,\\,q)=1}}f(n)e({a\\bar{n}\\over q})\\ll\\sqrt{\\tau(q)\\over q}N\\log\\log(6N)+q^{{1\\over 4}+{\\epsilon\\over 2}}N^{1\\over 2}(\\log(6N))^{1\\over 2}+{N\\over \\sqrt{\\log\\log(6N)}},$$ where $\\bar{n}$ is the multiplicative inverse of $n$ such that $\\bar{n}n\\equiv 1\\,({\\rm mod}\\,q),\\,e(x)=\\exp(2\\pi ix),\\,\\tau(q)$ is the divisor function."},"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":"1401.4556","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-18T16:03:31Z","cross_cats_sorted":[],"title_canon_sha256":"b190995c0115b4066598956fadc88dbc3fc3616ddad2097114e6c2c0e1a91c66","abstract_canon_sha256":"56f4976145db05ee85cae84e8c3dd27ab955722790204fcae2e4a30c57df46b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:32.111063Z","signature_b64":"pUGlYTx6aSx/QtbDTQi0jhBEm1I/JfY/nLBDiHT7uy0IdHD6CL2JhYpj/3tQfUVcpwGj/Ee5q7nr8rOYo/JBCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49747eb420911a3d4de5390ca1a8cb277eb42b19cabfb3140e66283dc047a9f0","last_reissued_at":"2026-05-18T02:56:32.110478Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:32.110478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kloosterman 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":"2014-01-18T16:03:31Z","abstract_excerpt":"Let $f(n)$ be a multiplicative function satisfying $|f(n)|\\leq 1$, $q$ $(\\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\\,q)=1$. 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