In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character
χ
\chi
to squarefree modulus
q
q
, we prove the following upper bound:
|
∑
1
⩽
n
⩽
N
χ
(
n
)
|
⩽
c
q
log
q
,
\begin{align*} \left | \sum _{1 \leqslant n\leqslant N} \chi (n) \right |\leqslant c \sqrt {q} \log q, \end{align*}
where
c
=
1
/
(
2
π
2
)
+
o
(
1
)
c=1/(2\pi ^2)+o(1)
for even characters and
c
=
1
/
(
4
π
)
+
o
(
1
)
c=1/(4\pi )+o(1)
for odd characters, with an explicit
o
(
1
)
o(1)
term. This improves a result of Frolenkov and Soundararajan for large
q
q
. We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of
log
q
\log {q}
as in previous approaches and is an important factor for fully explicit bounds.