Theta function of a lattice

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

:$\backslash Theta\_\backslash Lambda(\backslash tau)\; =\; \backslash sum\_\{x\backslash in\backslash Lambda\}e^\{i\backslash pi\backslash tau\backslash |x\backslash |^2\}\backslash qquad\backslash mathrm\{Im\}\backslash ,\backslash tau\; >\; 0.$

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 $q\; =\; e^\{2i\backslash pi\backslash tau\}$ so that the coefficient of qⁿ gives the number of lattice vectors of norm 2n.

• Siegel theta series
• Theta constant

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

Category:Theta functions

{{numtheory-stub}}