Theorem
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":
formula_1
The standard way to prove it is to put τ =  + "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):
formula_2
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
formula_3
provided that we add the hypothesis that "pq" is an even number.