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