Landsberg-Schaar relation

The Landsberg-Schaar relation states that for any positive integers $p$and $q$:

$$\frac{1}{\sqrt{p}}\sum _{n=0}^{p-1}\exp \left(\frac{2\pi i{n}^{2}q}{p}\right)=\frac{e^{\pi i/4}}{\sqrt{2q}}\sum _{n=0}^{2q-1}\exp \left(-\frac{\pi i{n}^{2}p}{2q}\right)$$(1)

Although both sides of (1) are mere finite sums,no one has yet found a proof which uses no infinitelimiting process. One way to prove it is to put$\tau =2iq/p+ϵ$, where $ϵ>0$, inthis identity due to Jacobi:

$$\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{-\pi n^2\tau }=\frac{1}{\sqrt{\tau }}\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{-\pi n^2/\tau }$$(2)

and let $ϵ\to 0$. The details can be found here (http://planetmath.org/ProofOfJacobisIdentityForVarthetaFunctions). The identity (2) is a basic one in the theory oftheta 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 formulafor 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.