Theta Series
One can associate to any (positive-definite) lattice Λ a theta function given by
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 so that the coefficient of qn gives the number of lattice vectors of norm 2n. In the Leech lattice, there are 196560 vectors of norm 4, 16773120 vectors of norm 6, 398034000 vectors of norm 8 and so on. The theta series of the Leech lattice is thus:
where represents the Ramanujan tau function, and is the divisor function. It follows that the number of vectors of norm 2m is
Read more about this topic: Leech Lattice
Famous quotes containing the word series:
“As Cuvier could correctly describe a whole animal by the contemplation of a single bone, so the observer who has thoroughly understood one link in a series of incidents should be able to accurately state all the other ones, both before and after.”
—Sir Arthur Conan Doyle (1859–1930)