Landsberg–Schaar relation - Alchetron, the free social encyclopedia

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:

1 p ∑ n = 0 p − 1 exp ⁡ ( 2 π i n 2 q p ) = e 1 4 π i 2 q ∑ n = 0 2 q − 1 exp ⁡ ( − π i n 2 p 2 q ) .

Although both sides are mere finite sums, no proof by entirely finite methods has yet been found. The standard way to prove it 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):

∑ n = − ∞ + ∞ e − π n 2 τ = 1 τ ∑ n = − ∞ + ∞ e − π n 2 τ

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

1 p ∑ n = 0 p − 1 exp ⁡ ( π i n 2 q p ) = e 1 4 π i q ∑ n = 0 q − 1 exp ⁡ ( − π i n 2 p q )

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

References

Landsberg–Schaar relation Wikipedia

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