Averages of character sums

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 $\approx aq\mathrm{/}b$ is $\gg q^{1/2} \log \log q$forall$a$relativelyprimeto$b$; another is that if the least quadratic nonresidue modulo $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ is large, then the character sum gets as large as $(\sqrt{q}/\pi) (L(1, \chi) + \log 2 - \epsilon)$,andif$B$isthisnonresidue,thenthereisasumoflength$q/B$whichhassize$(\sqrt{q}/\pi) (\log 2 - \epsilon)$.