# Gauss sum variant: $\sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)} \big)$

Motivated by an example in Chern Simons theory, let $p \in \mathbb{Z}$ be prime, can anyone compute this sum:

$$\sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)} \big)$$

This looks like the Gauss sum, except it is multiplied by something that is not periodic so we sum over $a, b \in \mathbb{Z}$.

• $2\cosh(\pi(a-b))$ grows quite fast as $|a-b|\to +\infty$. Such series is simply not convergent. – Jack D'Aurizio Jan 9 '17 at 18:07