Theta function of a lattice explained

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

\Theta _{Λ(\tau)} = \sum _{x\inΛ} e ^{i\pi\tau\|x\| ^{2}} Im \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\pi\tau} so that the coefficient of qⁿ gives the number of lattice vectors of norm 2n.

See also

• Siegel theta series
• Theta constant