# 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

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

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

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

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:

 $K_{s}(a)=\frac{1}{2}\int_{0}^{\infty}x^{s-1}\exp\left(\frac{-a(x+x^{-1})}{2}% \right)dx$

where $\Re(a)>0$.

Title Kloosterman sum KloostermanSum 2013-03-22 13:59:33 2013-03-22 13:59:33 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 11L05 msc 43A25 GaussSum