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:

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

The standard way to prove it[1] is to put τ = 2iq/p + ε, where ε > 0 in this identity due to Jacobi (which is essentially just a special case of the Poisson summation formula in classical harmonic analysis):

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

and then let ε → 0.

A proof[2] using only finite methods was discovered in 2018 by Ben Moore.

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

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

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

References

1. Dym, H.; McKean, H. P. (1972). Fourier Series and Integrals. Academic Press. ISBN 978-0122264511.
2. Moore, Ben (2019-07-17). "A proof of the Landsberg-Schaar relation by finite methods". arXiv:1810.06172 [math.NT].