Gauss sum

The Gaussian sum or Gaussian sum is a specific type of a finite sum of roots of unity, typically

The sum goes over the elements r of a finite commutative ring R, ψ (r ) is a group homomorphism of the abelian group R in the unit circle and χ (r ) is a group homomorphism of the unit group R × in the unit circle, continued ( by the value 0 ) on non r units. Gaußsummen are analogies to the finite fields of the gamma function. Such sums are ubiquitous in number theory. See, for example, Use in the functional equations of Dirichlet L- function, where a Dirichlet character χ the equation in the relationship between L (s, χ ) and L ( 1 - s, χ *) the factor

Used, χ * is the complex conjugate of χ.

Originally looked at Carl Friedrich Gauss, the quadratic Gaussian sum for a field R of residuals modulo a prime p and χ the Legendre symbol. Gauss showed that G ( χ ) = p1 / 2 or ip1 / 2, depending on whether p is congruent to 1 modulo 4, or 3.

An alternative form of the Gaussian sum is:

Quadratic Gaussian sums are closely connected with the theory of theta-functions.

The general theory of Gaussian sum was in the early nineteenth century, using the Jacobi sums and their prime factorization, developed in cyclotomic bodies. Buzz about the quantities, where χ takes on a particular value when the underlying ring of the residuals ring modulo an integer N are described by the theory of Gaussian periods.

The absolute value of a Gaussian sum is usually used as an application of the theorem of Plancherel on finite groups. In the case that R is a body of p elements, and χ is not trivial, is the absolute value of P1 / 2 The determination of the exact value of general Gaussian sums from the result of Gaussian for the square case is a long unsolved problem. For some cases, see Kummer sum.