Benson's Formula

An equation for a lattice sum  (Borwein and Bailey 2003, p. 26)

			(1)
			(2)

Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to  (OEIS A085469), a value known as "the" Madelung constant.

No closed form for  is known (Bailey et al. 2006).

REFERENCES:

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.

Borwein, J.; Bailey, D.; and Girgensohn, R. §4.3.2 and 4.3.3 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 301, 1987.

Finch, S. R. "Madelung's Constant." §1.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 76-81, 2003.

Sloane, N. J. A. Sequence A085469 in "The On-Line Encyclopedia of Integer Sequences."