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Lattice Sum

Cubic lattice sums include the following:

$\displaystyle b_2(2s)$ $\textstyle \equiv$ $\displaystyle \setbox0=\hbox{$\scriptstyle{i, j=-\infty}$}\setbox2=\hbox{$\disp...
...\mathop{{\sum}'}_{\kern-\wd4 i, j=-\infty}^\infty {(-1)^{i+j}\over (i^2+j^2)^s}$ (1)
$\displaystyle b_3(2s)$ $\textstyle \equiv$ $\displaystyle \setbox0=\hbox{$\scriptstyle{i, j, k=-\infty}$}\setbox2=\hbox{$\d...
...\sum}'}_{\kern-\wd4 i, j, k=-\infty}^\infty {(-1)^{i+j+k}\over (i^2+j^2+k^2)^s}$ (2)
$\displaystyle b_n(2s)$ $\textstyle \equiv$ $\displaystyle \setbox0=\hbox{$\scriptstyle{k_1, \ldots, k_n=-\infty}$}\setbox2=...
...s, k_n=-\infty}^\infty
{(-1)^{k_1+\ldots+k_n}\over ({k_1}^2+\ldots+{k_n}^2)^s},$ (3)

where the prime indicates that summation over $(0, 0, 0)$ is excluded. As shown in Borwein and Borwein (1987, pp. 288-301), these have closed forms for even $n$
$\displaystyle b_2(2s)$ $\textstyle =$ $\displaystyle -4\beta(s)\eta(s)$ (4)
$\displaystyle b_4(2s)$ $\textstyle =$ $\displaystyle -8\eta(s)\eta(s-1)$ (5)
$\displaystyle b_8(2s)$ $\textstyle =$ $\displaystyle -16\zeta(s)\eta(s-3),\quad {\rm for\ } \Re[s]>1$ (6)

where $\beta(z)$ is the Dirichlet Beta Function, $\eta(z)$ is the Dirichlet Eta Function, and $\zeta(z)$ is the Riemann Zeta Function. The lattice sums evaluated at $s=1$ are called the Madelung Constants. Borwein and Borwein (1986) prove that $b_8(2)$ converges (the closed form for $b_8(2s)$ above does not apply for $s=1$), but its value has not been computed.

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

h_2(2s)\equiv {4\over 3}\sum_{m, n=-\infty}^\infty {\sin[(n+...
...tstyle{1\over 2}}m)^2+3({\textstyle{1\over 2}}m)^2}\right]^s},
\end{displaymath} (7)

where $\theta \equiv 2\pi/3$. This Madelung Constant is expressible in closed form for $s=1$ as
h_2(2)=\pi\ln 3\sqrt{3}\,.
\end{displaymath} (8)

See also Benson's Formula, Madelung Constants


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.''

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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© 1996-9 Eric W. Weisstein