# Landsberg-Schaar relation

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

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

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:

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

and let $\epsilon\to 0$. The details can be found here (http://planetmath.org/ProofOfJacobisIdentityForVarthetaFunctions). 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.

Title Landsberg-Schaar relation LandsbergSchaarRelation 2013-03-22 13:23:20 2013-03-22 13:23:20 mathcam (2727) mathcam (2727) 8 mathcam (2727) Theorem msc 11L05 Schaar’s identity RiemannThetaFunction