The Landsberg-Schaar relation states that for any positive integers $p$ and $q$ : \begin{equation} \frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi in^2q}{p}\right)= \frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi in^2p}{2q}\right) \end{equation}Although both sides of (1) are mere finite sums, no one has yet found a proof which uses no infinite limiting process. One way to prove it is to put $\tau=2iq/p+\epsilon$ , where $\epsilon>0$ , in this identity due to Jacobi: \begin{equation} \sum_{n=-\infty}^{+\infty}e^{-\pi n^2\tau}=\frac{1}{\sqrt{\tau}} \sum_{n=-\infty}^{+\infty}e^{-\pi n^2/\tau} \end{equation} and let $\epsilon\to 0$ . The details can be found here. The identity (2) is a basic one in the theory of theta functions. It is sometimes called the functional equation for the Riemann theta function. See e.g. [2 VII.6.2].

If we just let $q=1$ in the Landsberg-Schaar identity, it reduces to a formula for the quadratic Gauss sum mod $p$ ; notice that $p$ need not be prime.

References:

[1] H. Dym and H.P. McKean. Fourier Series and Integrals. Academic Press, 1972.

[2] J.-P. Serre. A Course in Arithmetic. Springer, 1970.