 TOPICS  # Theta Series

The theta series of a lattice is the generating function for the number of vectors with norm in the lattice.

Theta series for a number of lattices are implemented in the Wolfram Language as LatticeData[lattice, "ThetaSeriesFunction"].

The following table summarized lattice with closed-form theta series. Here, is a Jacobi theta function.

 lattice theta series generating function Barnes-Wall lattice body-centered cubic lattice Coxeter-Todd lattice face-centered cubic lattice hexagonal close packing lattice hexagonal lattice Leech lattice simple cubic lattice square lattice tetrahedral packing lattice The following tables gives the first few terms of the series for these 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 Coxeter-Todd Lattice, Eisenstein Series, Leech Lattice

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## References

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, 1993.Sloane, N. J. A. Sequences A004010/M5478, A004013/M4473, A004015/M4821, A004018/M3218, A005875/M4092, A008408, and A008409 in "The On-Line Encyclopedia of Integer Sequences."

Theta Series

## Cite this as:

Weisstein, Eric W. "Theta Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThetaSeries.html