Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
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. Gauss sums are the analogues for finite fields of the Gamma function.
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) (where χ is the complex conjugate of χ) involves a factor
The case originally considered by Carl Friedrich 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 Gauss proved that G(χ) = p 1⁄2 or ip 1⁄2 for p congruent to 1 or 3 modulo 4 respectively.
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