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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. 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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. 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