Gauss sum

In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically

${\displaystyle G(\chi ):=G(\chi ,\psi )=\sum \chi (r)\cdot \psi (r)}$

where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R⁺ into the unit circle, and χ 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.^[1]

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^{[clarification needed]}

${\displaystyle {\frac {G(\chi )}{|G(\chi )|}}.}$