Kloosterman sum

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

$$K_p(a,b)=\sum _{x\in {𝔽}_p^{*}}\exp \left(\frac{2\pi i(ax+b{x}^{-1})}{p}\right)$$

where ${𝔽}_p$ 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, ${𝔽}_q$ the field of $q$ elements, $\chi :{𝔽}_q^{*}\to ℂ$ a character, and $\psi :{𝔽}_q\to ℂ$ a mapping such that $\psi (x+y)=\psi (x)\psi (y)$ identically. The sums

$$K_{\psi }(\chi |a,b)=\sum _{x\in {𝔽}_q^{*}}\chi (x)\psi (ax+b{x}^{-1})$$

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

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

$$K_s(a)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{x}^{s-1}\exp \left(\frac{-a(x+x^{-1})}{2}\right)𝑑x$$

where $\mathrm{\Re }(a)>0$.

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