Character sum


In mathematics, a character sum is a sum
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.
Assume χ is a nonprincipal Dirichlet character to the modulus N.

Sums over ranges

The sum taken over all residue classes mod N is then zero. This means that the cases of interest will be sums over relatively short ranges, of length R < N say,
A fundamental improvement on the trivial estimate is the Pólya–Vinogradov inequality, stating in big O notation that
Assuming the generalized Riemann hypothesis, Hugh Montgomery and R. C. Vaughan have shown that there is the further improvement

Summing polynomials

Another significant type of character sum is that formed by
for some function F, generally a polynomial. A classical result is the case of a quadratic, for example,
and χ a Legendre symbol. Here the sum can be evaluated, a result that is connected to the local zeta-function of a conic section.
More generally, such sums for the Jacobi symbol relate to local zeta-functions of elliptic curves and hyperelliptic curves; this means that by means of André Weil's results, for N = p a prime number, there are non-trivial bounds
The constant implicit in the notation is linear in the genus of the curve in question, and so can be taken as the degree of F.
Weil's results also led to the Burgess bound, applying to give non-trivial results beyond Pólya–Vinogradov, for R a power of N greater than 1/4.
Assume the modulus N is a prime.
for any integer r ≥ 3.