Landsberg–Schaar relation explained

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:

(1)/(\sqrt{p) }\sum_^\exp\left(\frac\right)=\frac\sum_^\exp\left(-\frac\right).

The standard way to prove it^[1] is to put  =  + 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):

\sum ^{+inf ty} _{n =-inf ty} e ^{- \pi n ^{2\tau}} = (1)/(\sqrt{\tau) }\sum_^e^

and then let ε → 0.

A proof using only finite methods was discovered in 2018 by Ben Moore.^{[2] [3]}

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

(1)/(\sqrt{p) }\sum_^\exp\left(\frac\right)=\frac\sum_^\exp\left(-\frac\right)

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

References

1. Book: H.. Dym. H. P.. McKean. Fourier Series and Integrals. Academic Press. 1972 . 978-0122264511.
2. Moore. Ben. 2020-12-01. A proof of the Landsberg–Schaar relation by finite methods. The Ramanujan Journal. en. 53. 3. 653–665. 10.1007/s11139-019-00195-4. 1810.06172. 55876453. 1572-9303.
3. Moore. Ben. 2019-07-17. A proof of the Landsberg-Schaar relation by finite methods. math.NT. 1810.06172.