Character sum

Summary

A character sum is a sum of the values of a Dirichlet character over a range of integers, such as $\sum_{M < n \le M + N} \chi(n)$. Over a full period the sum is $0$ for a nonprincipal character. So the interest lies in incomplete sums and in sums of $\chi(f(n))$ for polynomials $f$. Bounds for character sums control the distribution of quadratic residues and of primitive roots, and they enter the study of Dirichlet L-functions.