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

${\mathrm{\Theta }}_{\mathrm{\Lambda }}(\tau )=\sum _{x\in \mathrm{\Lambda }}{e}^{i\pi \tau \Vert x{\Vert }^2}\mathrm{I}\mathrm{m}\tau >0.$

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.

References

• Deconinck, Bernard (2010), "Multidimensional Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248