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

is excluded. As shown in Borwein and Borwein (1987, pp. 288-301), these have closed forms for even

$\displaystyle b_2(2s)$	$\textstyle =$	$\displaystyle -4\beta(s)\eta(s)$	(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

are called the Madelung Constants. Borwein and Borwein (1986) prove that

converges (the closed form for

above does not apply for

), but its value has not been computed.

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

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

(7)

where $\theta \equiv 2\pi/3$ . This Madelung Constant is expressible in closed form for

$\begin{displaymath} h_2(2)=\pi\ln 3\sqrt{3}\,. \end{displaymath}$

(8)

See also Benson's Formula, Madelung Constants

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.