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
Θ Λ ( τ ) = ∑ x ∈ Λ e i π τ ‖ x ‖ 2 I m τ > 0. {\displaystyle \Theta _{\Lambda }(\tau )=\sum _{x\in \Lambda }e^{i\pi \tau \|x\|^{2}}\qquad \mathrm {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 2 i π τ {\displaystyle 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
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.