Theta function of a lattice

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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

ΘΛ(τ)=xΛeiπτx2Imτ>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=e2iπτ 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