Cubic lattice sums include the following:

(1) | |||

(2) | |||

(3) |

where the prime indicates that summation over is excluded. As shown in Borwein and Borwein (1987, pp. 288-301), these have closed forms for even

(4) | |||

(5) | |||

(6) |

where is the Dirichlet Beta Function, is the Dirichlet Eta Function, and is the Riemann Zeta Function. The lattice sums evaluated at are called the Madelung Constants. Borwein and Borwein (1986) prove that converges (the closed form for above does not apply for ), but its value has not been computed.

For hexagonal sums, Borwein and Borwein (1987, p. 292) give

(7) |

(8) |

**References**

Borwein, D. and Borwein, J. M. ``On Some Trigonometric and Exponential Lattice Sums.'' *J. Math. Anal.* **188**, 209-218, 1994.

Borwein, D.; Borwein, J. M.; and Shail, R. ``Analysis of Certain Lattice Sums.'' *J. Math. Anal.* **143**, 126-137, 1989.

Borwein, D. and Borwein, J. M. ``A Note on Alternating Series in Several Dimensions.'' *Amer. Math. Monthly* **93**, 531-539, 1986.

Borwein, J. M. and Borwein, P. B. *Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.*
New York: Wiley, 1987.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/mdlung/mdlung.html

Glasser, M. L. and Zucker, I. J. ``Lattice Sums.'' In *Perspectives in Theoretical Chemistry: Advances and Perspectives* **5**,
67-139, 1980.

© 1996-9

1999-05-26