Landsberg-Schaar relation
The Landsberg-Schaar relation states that for any positive integers and :
(1) |
Although both sides of (1) are mere finite sums,
no one has yet found a proof which uses no infinite
limiting process. One way to prove it is to put
, where , in
this identity
due to Jacobi:
(2) |
and let . The details can be found here (http://planetmath.org/ProofOfJacobisIdentityForVarthetaFunctions). The identity (2) is a basic one in the theory of
theta functions. It is sometimes called the functional equation for the Riemann theta function
. See e.g. [2 VII.6.2].
If we just let in the Landsberg-Schaar identity, it reduces to a formula
for the quadratic Gauss sum mod ; notice that need not be prime.
References:
[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 |