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# 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 $\Re(a)>0$.

Related:

GaussSum

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

11L05*no label found*43A25

*no label found*

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