For a primitive Dirichlet character
χ
\chi
modulo
q
q
, we define
M
(
χ
)
=
max
t
|
∑
n
≤
t
χ
(
n
)
|
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
≥
3
g\geq 3
. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character
χ
\chi
we have
M
(
χ
)
≪
ε
q
(
log
q
)
1
−
δ
g
(
log
log
q
)
−
1
/
4
+
ε
,
\begin{equation*} M(\chi )\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*}
where
δ
g
≔
1
−
g
π
sin
(
π
/
g
)
\delta _g ≔1-\frac {g}{\pi }\sin (\pi /g)
. This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that
M
(
χ
)
≪
q
(
log
2
q
)
1
−
δ
g
(
log
3
q
)
−
1
4
(
log
4
q
)
O
(
1
)
,
\begin{equation*} M(\chi ) \ll \sqrt {q} \left (\log _2 q\right )^{1-\delta _g} \left (\log _3 q\right )^{-\frac {1}{4}}\left (\log _4 q\right )^{O(1)}, \end{equation*}
where
log
j
\log _j
is the
j
j
-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of
log
4
q
\log _4 q
). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.