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 $q\equiv 3(\mathrm{mod}4)$ 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 $(\sqrt{q}/\pi )(\log 2-\epsilon )$.

Cited by in corpus (2)

• The frequency and the structure of large character sums
• Large values of $L(1,\chi )$ for $k$-th order characters $\chi$ and applications to character sums