The theta series of a point lattice 
is the generating function
 |
for 
, where 
is the Euclidean vector norm.
The coefficient of 
counts the vectors with squared
vector norm 
in the point lattice.
Theta series for a number of point lattices are implemented in the Wolfram Language as LatticeData[lattice, "ThetaSeriesFunction"].
The following table summarizes point lattices with closed-form theta series. Here, 
is a Jacobi
theta function.
| lattice | theta series generating function |
| Barnes-Wall lattice | ![1/2[theta_2^(16)(q)+theta_3^(16)(q)+theta_4^(16)(q)+30theta_2^8(q)theta_3^8(q)]](/images/equations/ThetaSeries/Inline7.svg) |
| body-centered cubic lattice |  |
| Coxeter-Todd lattice | ![9/(32)theta_2^6(q)theta_2(q^3)^6+[theta_2(q^4)theta_2(q^(12))+theta_3(q^4)theta_3(q^(12))]^6+(45)/(16)theta_2^4(q)[theta_2(q^4)theta_2(q^(12))+theta_3(q^4)theta_3(q^(12))]^2theta_2^4(q^3)](/images/equations/ThetaSeries/Inline9.svg) |
| face-centered cubic lattice | ![1/2[theta_3^3(q)+theta_4^3(q)]](/images/equations/ThetaSeries/Inline10.svg) |
| hexagonal close packing lattice | ![1/2theta_2(q^(8/3))[theta_2(q^(2/3))theta_2(q^2)+theta_3(q^(2/3))theta_3(q^2)]+[theta_3(q^(8/3))-1/2theta_2(q^(8/3))](theta_2(q^2)theta_2(q^6)+theta_3(q^2)theta_3(q^6))](/images/equations/ThetaSeries/Inline11.svg) |
| hexagonal lattice |  |
| Leech lattice | ![1/8[theta_2^8(q)+theta_3^8(q)+theta_4^8(q)]^3-(45)/(16)theta_2^8(q)theta_3^8(q)theta_4^8(q)](/images/equations/ThetaSeries/Inline13.svg) |
| simple cubic lattice |  |
| square lattice |  |
| tetrahedral packing lattice | ![1/2[theta_2^3(q)+theta_3^3(q)+theta_4^3(q)]](/images/equations/ThetaSeries/Inline16.svg) |
The following table gives the first few terms of the series for these point lattices.
| lattice | OEIS | theta series |
| Barnes-Wall lattice | A008409 |  |
| body-centered cubic lattice | A004013 |  |
| Coxeter-Todd lattice | A004010 |  |
| face-centered cubic lattice | A004015 |  |
| hexagonal close packing lattice |  | |
| hexagonal lattice |  | |
| Leech lattice | A008408 |  |
| simple cubic lattice | A005875 |  |
| square lattice | A004018 |  |
| tetrahedral packing lattice |  |
Writing 
,
for a point lattice 
with covolume 1, Luo
and Wei (2026) proved that 
is minimized for every 
uniquely, up to an orthogonal
transformation, by the rescaled D4
lattice 
.
Their computer-assisted proof uses certified interval bounds. Luo and Wei (2026) state that ChatGPT assisted with
review of proof arguments and citations and with preparation
of the computational verification materials.
