Character sum

Assume χ is a non-principal Dirichlet character to the modulus N. The sum taken over all residue classes mod N is then zero.

over relatively short ranges, of length R < N say, A fundamental improvement on the trivial estimate

is the Pólya–Vinogradov inequality, established independently by George Pólya and I. M. Vinogradov in 1918,[1][2] stating in big O notation that Assuming the generalized Riemann hypothesis, Hugh Montgomery and R. C. Vaughan have shown[3] that there is the further improvement Another significant type of character sum is that formed by for some function F, generally a polynomial.

Here the sum can be evaluated (as −1), 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 (Legendre symbol or hyperelliptic case) can be taken as the degree of F. (More general results, for other values of N, can be obtained starting from there.)