Kloosterman sum

Summary

A Kloosterman sum is an exponential sum over the invertible residues $x$ modulo $m$ with phase $(ax + b\bar{x})/m$. Here $\bar{x}$ is the inverse of $x$ modulo $m$. It is a real number, and for a prime modulus $p$ not dividing $ab$ Weil's bound gives $\abs{K(a, b; p)} \le 2\sqrt{p}$. Kloosterman sums appear in the Fourier coefficients of Poincaré series, in the circle method for quadratic forms, and in the spectral theory of automorphic forms.