  We show that a short truncation of the Fourier expansion for a character sum
gives a good approximation for the average value of that character sum over an
interval.
  We give a few applications of this result. One is that for any $b$ there are
infinitely many characters for which the sum up to $\approx aq/b$ is $\gg
q^{1/2} \log \log q$ for all $a$ relatively prime to $b$; another is that if
the least quadratic nonresidue modulo $q \equiv 3 \pmod 4$ is large, then the
character sum gets as large as $(\sqrt{q}/\pi) (L(1, \chi) + \log 2 -
\epsilon)$, and if $B$ is this nonresidue, then there is a sum of length $q/B$
which has size $(\sqrt{q}/\pi) (\log 2 - \epsilon)$.

Number Theory