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 same definition applies to the centers of a periodic sphere packing, after choosing a center as the origin.
The following table summarizes point lattices and nonlattice sphere packings with closed-form theta
series. Here, 
is a Jacobi theta function.
| point set | 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 | ![1/2theta_2(q^(16/3))[theta_2(q^(2/3))theta_2(q^2)+theta_3(q^(2/3))theta_3(q^2)]+[theta_3(q^(16/3))-1/2theta_2(q^(16/3))](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 | ![1/2[theta_2^3(q)+theta_3^3(q)+theta_4^3(q)]](/images/equations/ThetaSeries/Inline16.svg) |
In the hexagonal close packing row, the nearest-neighbor squared distance is 2, and the vertical period squared length is 16/3 (Sloane 1981, pp. 333-334).
The following table gives the first few terms of the series for these structures.
| point set | OEIS | theta series |
| Barnes-Wall lattice | A008409 |  |
| body-centered cubic lattice | A004013 |  |
| Coxeter-Todd lattice | A004010 |  |
| face-centered cubic lattice | A004015 |  |
| hexagonal close packing |  | |
| hexagonal lattice |  | |
| Leech lattice | A008408 |  |
| simple cubic lattice | A005875 |  |
| square lattice | A004018 |  |
| tetrahedral packing |  |
The coefficients are indexed by powers of 
in OEIS A004013, OEIS
A005875, and OEIS A004018,
and by powers of 
in OEIS A004010, OEIS A004015,
and OEIS A008408. In OEIS A008409,
the index is the power of 
in the normalization used here.
Writing 
,
for a point lattice 
containing the origin
with covolume 1, Luo and Wei (2026, p. 1) 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, p. 35) state that ChatGPT assisted with review of proof
arguments and with preparation of the computational verification materials.
