Landsberg–Schaar relation

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.

References

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  1. ^ Dym, H.; McKean, H. P. (1972). Fourier Series and Integrals. Academic Press. ISBN 978-0122264511.
  2. ^ Moore, Ben (2020-12-01). "A proof of the Landsberg–Schaar relation by finite methods". The Ramanujan Journal. 53 (3): 653–665. arXiv:1810.06172. doi:10.1007/s11139-019-00195-4. ISSN 1572-9303. S2CID 55876453.
  3. ^ Moore, Ben (2019-07-17). "A proof of the Landsberg-Schaar relation by finite methods". arXiv:1810.06172 [math.NT].