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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'': | |||
:<math> | |||
\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi in^2q}{p}\right)= | |||
\frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi in^2p}{2q}\right). | |||
</math> | |||
Although both sides are mere finite sums, no proof by entirely finite methods has yet been found. The standard way to prove it<ref name="DymAndMcKean">H. Dym and H.P. McKean. ''Fourier Series and Integrals''. Academic Press, 1972.</ref> is to put <math>\tau=2iq/p+\varepsilon</math>, where <math>\varepsilon>0</math> in this identity due to [[Carl Gustav Jacob Jacobi|Jacobi]] (which is essentially just a special case of the [[Poisson summation formula]] in classical harmonic analysis): | |||
:<math> | |||
\sum_{n=-\infty}^{+\infty}e^{-\pi n^2\tau}=\frac{1}{\sqrt{\tau}} | |||
\sum_{n=-\infty}^{+\infty}e^{-\pi n^2/\tau} | |||
</math> | |||
and then let <math>\varepsilon\to 0.</math> | |||
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 | |||
:<math> | |||
\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{\pi in^2q}{p}\right)= | |||
\frac{e^{\pi i/4}}{\sqrt{q}}\sum_{n=0}^{q-1}\exp\left(-\frac{\pi in^2p}{q}\right) | |||
</math> | |||
provided that we add the hypothesis that ''pq'' is an even number. | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Landsberg-Schaar relation}} | |||
[[Category:Analytic number theory]] | |||
[[Category:Theorems in number theory]] |
Revision as of 16:18, 26 February 2013
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:
Although both sides are mere finite sums, no proof by entirely finite methods has yet been found. The standard way to prove it[1] is to put , where in this identity due to Jacobi (which is essentially just a special case of the Poisson summation formula in classical harmonic analysis):
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
- ↑ H. Dym and H.P. McKean. Fourier Series and Integrals. Academic Press, 1972.