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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.
==Definition==
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:<math>\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0.</math>
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''.
== See also ==
* [[Siegel theta series]]
* [[Theta constant]]
==References==
*{{dlmf|id=21|title=Multidimensional Theta Functions|first=Bernard |last=Deconinck}}
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{{numtheory-stub}}
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