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

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

References

↑ H. Dym and H.P. McKean. Fourier Series and Integrals. Academic Press, 1972.

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