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Benson's Formula

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Benson's formula expresses the conditionally convergent lattice sum b_3(1) as an exponentially convergent series (Borwein and Bailey 2003, p. 26),

b_3(1)=lim_(N->infty)sum^'_(i,j,k=-N)^N((-1)^(i+j+k))/(sqrt(i^2+j^2+k^2))
(1)
=-12pisum_(m,n=1,3,...)^(infty)sech^2(1/2pisqrt(m^2+n^2)).
(2)

Here, N runs through the positive integers, the prime excludes (i,j,k)=(0,0,0), and sech is the hyperbolic secant. The limit uses expanding cubes centered at the origin. The corresponding sums over expanding spheres do not converge (Bailey et al. 2006). The value is -1.7475645946... and is known as "the" Madelung constant. The decimal expansion of its negation -b_3(1) is OEIS A085469.

No closed form for b_3(1) is known (Bailey et al. 2006).

See also

Lattice Sum, Madelung Constants

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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."

Referenced on Wolfram|Alpha

Benson's Formula

Cite this as:

Weisstein, Eric W. "Benson's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BensonsFormula.html

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