{"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"}