Editing Leech lattice (section)

==Theta series==

One can associate to any (positive-definite) lattice Λ a [[theta function]] given by

:<math>\Theta_\Lambda(\tau) = \sum_{x\in\Lambda} e^{i\pi\tau\|x\|^2} \qquad \operatorname{Im} \tau > 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 for the full [[modular group]] PSL(2,'''Z'''). The theta function of an integral lattice is often written as a [[power series]] in <math>q = e^{2i\pi\tau}</math> so that the coefficient of ''q''<sup>''n''</sup> gives the number of lattice vectors of squared norm 2''n''. In the Leech lattice, there are 196560 vectors of squared norm 4, 16773120 vectors of squared norm 6, 398034000 vectors of squared norm 8 and so on. The theta series of the Leech lattice is

: <math>
\begin{align}
\Theta_{\Lambda_{24}}(\tau) & = E_{12}(\tau)-\frac{65520}{691} \Delta(\tau) \\[5pt]
& = 1 + \sum_{m=1}^\infty \frac{65520}{691} \left(\sigma_{11} (m) - \tau (m) \right) q^m \\[5pt]
& = 1 + 196560q^2 + 16773120q^3 + 398034000q^4 + \cdots,
\end{align}
</math>

where <math>E_{12}(\tau)</math> is the normalized [[Eisenstein series]] of weight 12, <math>\Delta(\tau)</math> is the [[modular discriminant]], <math>\sigma_{11}(n)</math> is the [[divisor function]] for exponent 11, and <math>\tau(n)</math> is the [[Ramanujan tau function]]. It follows that for ''m''≥1 the number of vectors of squared norm 2''m'' is

: <math> \frac{65520}{691} \left(\sigma_{11} (m) - \tau (m) \right).</math>