Character sum

In mathematics, a character sum is a sum ${\textstyle \sum \chi (n)}$ of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform).

Assume χ is a non-principal Dirichlet character to the modulus N.