Theta function of a lattice

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

[math]\displaystyle{ \Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau \gt 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]\displaystyle{ q = e^{2i\pi\tau} }[/math] so that the coefficient of qⁿ gives the number of lattice vectors of norm 2n.

See also

• Theta constant

References

• Deconinck, Bernard (2010), "Multidimensional Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/21