{"paper":{"title":"Large odd order character sums and improvements of the P\\'{o}lya-Vinogradov inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander P. Mangerel, Youness Lamzouri","submitted_at":"2017-01-04T15:10:56Z","abstract_excerpt":"For a primitive Dirichlet character $\\chi$ modulo $q$, we define $M(\\chi)=\\max_{t } |\\sum_{n \\leq t} \\chi(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\\geq 3$. Our main result provides a further improvement of the classical P\\'{o}lya-Vinogradov inequality in this case. More specifically, we show that for any such character $\\chi$ we have $$M(\\chi)\\ll_{\\varepsilon} \\sqrt{q}(\\log q)^{1-\\delta_g}(\\log\\log q)^{-1/4+\\varepsilon},$$ where $\\delta_g := 1-\\frac{g}{\\pi}\\sin(\\pi/g)$. This improves upon the works of Granville and Soundararajan and of Goldmakher. Furth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.01042","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"}