Benson's Formula

An equation for a Lattice Sum with $n=3$

$$-b_3(1)=\setbox0=\hbox{$\scriptstyle{i,\, j,\, k=-\infty}$}\... ... sech}\nolimits ^2({\textstyle{1\over 2}}\pi\sqrt{m^2+n^2}\,).$$

Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to $-1.74756\ldots$, a value known as ``the'' Madelung Constant.

See also Madelung Constants

References

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