# 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

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

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, $\chi :{\mathbb{F}}_{q}^{*}\to \u2102$ a character, and $\psi :{\mathbb{F}}_{q}\to \u2102$ a mapping such that $\psi (x+y)=\psi (x)\psi (y)$ identically. The sums

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

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:

$${K}_{s}(a)=\frac{1}{2}{\int}_{0}^{\mathrm{\infty}}{x}^{s-1}\mathrm{exp}\left(\frac{-a(x+{x}^{-1})}{2}\right)\mathit{d}x$$ |

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 |