Character sum

Summing polynomials

Another significant type of character sum is that formed by

${\displaystyle \sum \chi (F(n))}$

for some function F, generally a polynomial. A classical result is the case of a quadratic, for example,

${\displaystyle F(n)=n(n+1)}$

and χ a Legendre symbol. 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

${\displaystyle O(\sqrt{p}).}$

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.)

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.

${\displaystyle \begin{array}{cc}\mathrm{\Sigma }& \ll p^{1/2}\log p,\\ \mathrm{\Sigma }& \ll 2R^{1/2}{p}^{3/16}\log p,\\ \mathrm{\Sigma }& \ll r{R}^{1-1/r}{p}^{(r+1)/4r^2}(\log p{)}^{1/2r}\end{array}}$

for any integer r ≥ 3.