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| | In mathematics, the '''theta function of a lattice''' is a function whose coefficients give the number of vectors of a given norm. |
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| | ==Definition== |
| | One can associate to any (positive-definite) [[Lattice (discrete subgroup)|lattice]] Λ a [[theta function]] given by |
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| | :<math>\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0.</math> |
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| | The theta function of a [[Lattice (discrete subgroup)|lattice]] is then a [[holomorphic function]] on the [[upper half-plane]]. Furthermore, the theta function of an even [[unimodular lattice]] of rank ''n'' is actually a [[modular form]] of weight ''n''/2. The theta function of an integral lattice is often written as a power series in <math>q = e^{2i\pi\tau}</math> so that the coefficient of ''q''<sup>''n''</sup> gives the number of lattice vectors of norm 2''n''. |
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| | ==References== |
| | *{{dlmf|id=21|title=Multidimensional Theta Functions|first=Bernard |last=Deconinck}} |
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| | [[Category:Number theory]] |
| | [[Category:Special functions]] |
Revision as of 17:45, 19 October 2013
In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm.
Definition
One can associate to any (positive-definite) lattice Λ a theta function given by
The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2. The theta function of an integral lattice is often written as a power series in 
so that the coefficient of qn gives the number of lattice vectors of norm 2n.
References