PlanetMath: Landsberg-Schaar relation

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Landsberg-Schaar relation

(Theorem)

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

$\displaystyle \frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi in^2q}{p}\... ...{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi in^2p}{2q}\right)$

(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 $\tau=2iq/p+\epsilon$ , where $\epsilon>0$ , in this identity due to Jacobi:

$\displaystyle \sum_{n=-\infty}^{+\infty}e^{-\pi n^2\tau}=\frac{1}{\sqrt{\tau}} \sum_{n=-\infty}^{+\infty}e^{-\pi n^2/\tau}$

(2)

and let $\epsilon\to 0$ . The details can be found here. 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.

"Landsberg-Schaar relation" is owned by mathcam. [ full author list (2) | owner history (1) ]

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See Also: Riemann $þeta$

-function

Other names:

Schaar's identity

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Cross-references: Fourier series, references, prime, Gauss sum, Riemann theta function, functional equation, infinite, proof, sums, finite, sides, integers, positive
There is 1 reference to this entry.

This is version 5 of Landsberg-Schaar relation, born on 2003-01-25, modified 2006-07-24.
Object id is 3926, canonical name is LandsbergSchaarRelation.
Accessed 2933 times total.

Classification:

AMS MSC:

11L05 (Number theory :: Exponential sums and character sums :: Gauss and Kloosterman sums; generalizations)

Pending Errata and Addenda

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