Kloosterman sum

The Kloosterman sumMathworldPlanetmath is one of various trigonometric sums that are useful in number theoryMathworldPlanetmathPlanetmath and, more generally, in finite harmonic analysis. The original Kloosterman sum is


where 𝔽p is the field of prime order p. Such sums have been generalized in a few different ways since their introduction in 1926. For instance, let q be a prime power, 𝔽q the field of q elements, χ:𝔽q* a character, and ψ:𝔽q a mapping such that ψ(x+y)=ψ(x)ψ(y) identically. The sums


are of interest, because they come up as Fourier coefficients of modular formsMathworldPlanetmath.

Kloosterman sums are finite analogs of the K-Bessel functionsDlmfMathworldPlanetmathPlanetmath of this kind:


where (a)>0.

Title Kloosterman sum
Canonical name KloostermanSum
Date of creation 2013-03-22 13:59:33
Last modified on 2013-03-22 13:59:33
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 11L05
Classification msc 43A25
Related topic GaussSum