Landsberg-Schaar relation

The Landsberg-Schaar relation states that for any positive integers p and q:

1pn=0p-1exp(2πin2qp)=eπi/42qn=02q-1exp(-πin2p2q) (1)

Although both sides of (1) are mere finite sums, no one has yet found a proof which uses no infiniteMathworldPlanetmath limiting process. One way to prove it is to put τ=2iq/p+ϵ, where ϵ>0, in this identityPlanetmathPlanetmath due to Jacobi:

n=-+e-πn2τ=1τn=-+e-πn2/τ (2)

and let ϵ0. The details can be found here ( The identity (2) is a basic one in the theory of theta functionsDlmfMathworld. It is sometimes called the functional equation for the Riemann theta functionDlmfDlmfMathworldPlanetmath. See e.g. [2 VII.6.2].

If we just let q=1 in the Landsberg-Schaar identity, it reduces to a formulaMathworldPlanetmathPlanetmath for the quadratic Gauss sum mod p; notice that p need not be prime.


[1] H. Dym and H.P. McKean. Fourier Series and Integrals. Academic Press, 1972.

[2] J.-P. Serre. A Course in Arithmetic. Springer, 1970.

Title Landsberg-Schaar relation
Canonical name LandsbergSchaarRelation
Date of creation 2013-03-22 13:23:20
Last modified on 2013-03-22 13:23:20
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Theorem
Classification msc 11L05
Synonym Schaar’s identity
Related topic RiemannThetaFunction