Exponential sum

Summary

An exponential sum is a finite sum of numbers $e^{2\pi i f(n)}$ on the unit circle, where $f$ is a real function on the integers. The trivial bound for such a sum is its number of terms, and the aim is to show cancellation, that is, a much smaller bound. Such bounds are central in analytic number theory: they control equidistribution modulo one, the zeros of zeta functions, and the circle method. Weyl sums, Gauss sums and Kloosterman sums are important special cases.