Gauss sum

Gauss sum

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically

:"G"(χ, ψ) = Σ χ("r")ψ("r")

where the sum is over elements "r" of some finite commutative ring "R", ψ("r") is a group homomorphism of the additive group "R"+ into the unit circle, and χ("r") is a group homomorphism of the unit group "R"× into the unit circle, extended to non-unit "r" where it takes the value 0.

Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet L-functions, where for a Dirichlet character χ the equation relating "L"("s", χ) and "L"(1 − "s", χ*) involves a factor

:"G"(χ)/|"G"(χ)|,

where χ* is the complex conjugate of χ.

The case originally considered by C. F. Gauss was the quadratic Gauss sum, for "R" the field of residues modulo a prime number "p", and χ the Legendre symbol. In this case the alternate form for "G" as a pure exponential sum is obtained, removing the character by adding the relation that the sum of the "p"-th roots of unity is 0. Quadratic Gauss sums are closely connected with the theory of theta-functions.

The general theory of Gauss sums was developed in the early nineteenth century, with the use of Jacobi sums and their prime decomposition in cyclotomic fields. Sums over the sets where χ takes on a particular value, when the underlying ring is the residue ring modulo an integer "N", are described by the theory of Gaussian periods.

The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. The determination of the exact value of Gauss sums, following the result of Gauss on the quadratic case, is a long-standing issue. For some cases see Kummer sum.

ee also

* Stickelberger's theorem
* Hasse-Davenport relation
* Chowla-Mordell theorem

References

*cite book | author = Ireland and Rosen | title = A Classical Introduction to Modern Number Theory | publisher = Springer-Verlag | year = 1990 | id=ISBN 0-387-97329-X


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Quadratic Gauss sum — For the general type of Gauss sums see Gaussian period, Gauss sumIn mathematics, quadratic Gauss sums are certain sums over exponential functions with quadratic argument. They are named after Carl Friedrich Gauss, who studied them… …   Wikipedia

  • Gauss–Bonnet theorem — The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). It is named …   Wikipedia

  • Gauss–Newton algorithm — The Gauss–Newton algorithm is a method used to solve non linear least squares problems. It can be seen as a modification of Newton s method for finding a minimum of a function. Unlike Newton s method, the Gauss–Newton algorithm can only be used… …   Wikipedia

  • Gauss circle problem — In mathematics, the Gauss circle problem is the problem of determining how many lattice points there are in a circle centred at the origin and with radius r . The first progress on a solution was made by Carl Friedrich Gauss, hence its name.The… …   Wikipedia

  • Gauss–Markov theorem — This article is not about Gauss–Markov processes. In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear model in which the errors have expectation zero and are uncorrelated and have… …   Wikipedia

  • Gauss's law — In physics, Gauss s law, also known as Gauss s flux theorem, is a law relating the distribution of electric charge to the resulting electric field. It is one of the four Maxwell s equations, which form the basis of classical electrodynamics, and… …   Wikipedia

  • Gauss , Karl Friedrich — (1777–1855) German mathematician Gauss came of a peasant background in Brunswick, Germany, and his extraordinary talent for mathematics showed itself at a very early age. By the age of three, he had discovered for himself enough arithmetic to be… …   Scientists

  • Gauss-Lucas theorem — In complex analysis, the Gauss Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P . The set of roots of a real or complex polynomial is a set of points in the complex plane. The… …   Wikipedia

  • Gauss–Codazzi equations — In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds. They also have applications for embedded …   Wikipedia

  • Gauss's Method —    , DEGAUSS    Karl Friedrich Gauss (1777 1855), German mathematician and scientist, was one of the three greatest mathematicians of who ever lived, the others being Archimedes and Newton. (Of course, Gauss lived about a century before Albert… …   Dictionary of eponyms

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”