Theta function of a lattice: Difference between revisions

Revision as of 02:35, 9 October 2013

Revision as of 14:53, 24 October 2013

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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]]{{dn|date=October 2013}} is then a [[holomorphic function]] on the [[upper half-plane]]. Furthermore, the theta function of an even [[unimodular lattice]] of rank&nbsp;''n'' is actually a [[modular form]] of weight&nbsp;''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&nbsp;2''n''.

==References==