Lattice Sum
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 
isthe Riemann Zeta Function. The lattice sums evaluated at 
are called the Madelung Constants. Borwein andBorwein (1986) prove that 
converges (the closed form for 
above does not apply for ), but its valuehas not been computed.For hexagonal sums, Borwein and Borwein (1987, p. 292) give
 | (7) |
. This Madelung Constant is expressible in closed form for as | (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.
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