{"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 $(a,\\,q)=1$. 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4556","kind":"arxiv","version":4},"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"}