Gauss sum

Remarks

The most common case is $F = \Z/p\Z$ for a prime $p$ and $\psi(a) = \zeta^a$ with $\zeta = e^{2\pi i/p}$; one then writes $g(\chi) = \sum_{a=1}^{p-1} \chi(a) \zeta^a$. If $\chi$ and $\psi$ are both nontrivial, then $\abs{g(\chi, \psi)} = \sqrt{q}$.

Example: let $p = 3$ and let $\chi$ be the Legendre symbol modulo $3$, so $\chi(1) = 1$ and $\chi(2) = -1$. Then $g(\chi) = \zeta - \zeta^2 = i\sqrt{3}$, and $g(\chi)^2 = -3$, in line with the general rule $g(\chi)^2 = \left(\frac{-1}{p}\right) p$ for the quadratic character.

Non-example: the sum $\sum_{a \in \Z/p\Z} \chi(a) e^{2\pi i a^2/p}$ is not a Gauss sum in this sense, because $a \mapsto e^{2\pi i a^2/p}$ is not additive, so it is not an additive character.

For a Dirichlet character $\chi$ modulo $n$ one defines in the same way $g(\chi) = \sum_{a \bmod n} \chi(a) e^{2\pi i a/n}$.