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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.
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.
References
^Dym, H.; McKean, H. P. (1972). Fourier Series and Integrals. Academic Press. ISBN978-0122264511.