Gauss sum (reviewed)

Let $\chi$ be a Dirichlet character with modulus $q$. For $a \in \mathbb{Z}$, the Gauss sum associated to $\chi$ and $a$ is given by $$ \large \tau_{a}(\chi) = \sum_{r \,\in\, \mathbb{Z}/q\mathbb{Z}} \chi(r)\, e^{2\pi iar/q}. $$ The Gauss sum $\tau_1(\chi)$ is often called "the Gauss sum" of $\chi$.