Landsberg–Schaar relation

Template:Short description Template:Use American English 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:

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

The standard way to prove it is to put Template:Mvar = Template:Sfrac + 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):

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

and then let ε → 0.

A proof 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

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

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