Theta function of a lattice: Difference between revisions

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In mathematics, the '''theta function of a lattice''' is a functionwhose coefficients give the nuber of vectors of a given norm.

#redirect[[E8 lattice#Theta function]]

==Definition==

One can associate to any (positive-definite) lattice Λ a [[theta function]] given by

:<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 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''^''n'' gives the number of lattice vectors of norm ''2n''.

==References==

*{{dlmf|id=21|title=Multidimensional Theta Functions|first=Bernard |last=Deconinck}}

[[Category:number theory]]

[[Category:special functions]]