I've read that the Kloosterman sum $$S(m, n; c) := \sum_{a \pmod c}^* e \left( \frac{m a + n \overline{a}}{c} \right)$$ is real for any $m, n \in \mathbf{Z}$ and $c \geq 1$. After a lot of scribbling I couldn't find a proof for this fact and on the web it is always stated without proof.
Can anyone help me on this?
Thanks!
That the Kloosterman sum is real is given as an exercise in the article at the following link: http://www.math.harvard.edu/~elkies/M259.02/kloos.pdf