Kloosterman sum (reviewed)

Let $\chi$ be a Dirichlet character with modulus $q$. The Kloosterman sum $K(a,b,\chi)$ is defined by \[K(a,b,\chi) = \sum_{r \in (\mathbb{Z}/q\mathbb{Z})^\times} \chi(r)\zeta^{ar+br^{-1}},\]

where $\zeta = e^{2\pi i/q}$. This reduces to the Gauss sum if $b=0$.