{"paper":{"title":"On the Fractional Parts of Polynomials Modulo $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chen Lin, Xuejun Guo, Zhefeng Xu","submitted_at":"2026-07-23T12:31:23Z","abstract_excerpt":"We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $\\varphi(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\\#\\left\\{1\\leq x< p/2:\\left\\{{\\varphi(x)}/{p}\\right\\}>\\frac12\\right\\} =\\frac{p}{4}+O_\\varphi(\\sqrt p\\log^2 p). $\nWe then show that the error term can be improved to $O_\\varphi(\\sqrt p\\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symme"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21259","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/2607.21259/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"}