Landsberg–Schaar relation

In number theory and harmonic analysis, the Landsberg–Schaar relation (or identity) is the following equation, which is valid for arbitrary positive integers p and q:

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

Although both sides are mere finite sums, no proof by entirely finite methods has yet been found. The standard way to prove it is to put $\tau =2iq/p+\varepsilon$, where $\varepsilon >0$ in this identity due to Jacobi (which is essentially just a special case of the Poisson summation formula in classical harmonic analysis):

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

and then let $\varepsilon \to 0$.

If we let q = 1, the identity reduces to a formula for the quadratic Gauss sum modulo p.

The Landsberg–Schaar identity can be rephrased more symmetrically as

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

provided that we add the hypothesis that pq is an even number.