Averages of character sums

Abstract: 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 $(\sqrt{q}/\pi) (\log 2 - \epsilon)$.