Lattice Sum
Cubic lattice sums include the following:
where the prime indicates that the origin (0,0), (0,0,0), etc. is excluded from the sum (Borwein and Borwein 1986, p. 288).
These have closed forms for even n,
for R[s]>1, where beta(z) is the Dirichlet beta function, eta(z) is the Dirichlet eta function, and zeta(z) is the Riemann zeta function (Zucker 1974, Borwein and Borwein 1987, pp. 288-301). The lattice sums evaluated at s=1 are called the Madelung constants. An additional form for b_2(2s) is given by
for R[s]>1/3, where r_2(n) is the sum of squares function, i.e., the number of representations of n by two squares (Borwein and Borwein 1986, p. 291). 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. A number of other related double series can be evaluated analytically.
For hexagonal sums, Borwein and Borwein (1987, p. 292) give
| h_2(2s)=4/3sum_(m,n=-infty)^infty(sin[(n+1)theta]sin[(m+1)theta]-sin(ntheta)sin[(m-1)theta])/([(n+1/2m)^2+3(1/2m)^2]^s), |
(9)
|
where theta=2pi/3. This Madelung constant is expressible in closed form for s=1 as
| h_2(2)=piln3sqrt(3). |
(10)
|
Other interesting analytic lattice sums are given by
| [画像: sum_(k,m,n=-infty)^infty((-1)^(k+m+n))/([(k+1/6)^2+(m+1/6)^2+(n+1/6)^2]^s) =12^sbeta(2s-1), ] |
(11)
|
![画像: sum_(k,m,n=-infty)^infty((-1)^(k+m+n))/([(k+1/6)^2+(m+1/6)^2+(n+1/6)^2]^s) =12^sbeta(2s-1),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LatticeSum/NumberedEquation4.svg){kind=link}
giving the special case
![画像: sum_(k,m,n=-infty)^infty((-1)^(k+m+n))/([(k+1/6)^2+(m+1/6)^2+(n+1/6)^2]^(1/2))=sqrt(3)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LatticeSum/NumberedEquation5.svg){kind=link}
(Borwein and Borwein 1986, p. 303), and
(Borwein and Borwein 1986, p. 305).
See also
Benson's Formula, Double Series, Epstein Zeta Function, Grenz-Formel, Madelung ConstantsExplore with Wolfram|Alpha
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References
Borwein, D. and Borwein, J. M. "A Note on Alternating Series in Several Dimensions." Amer. Math. Monthly 93, 531-539, 1986.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.; Borwein, J. M.; and Taylor, K. F. "Convergence of Lattice Sums and Madelung's Constant." J. Math. Phys. 26, 2999-3009, 1985.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. R. "Madelung's Constant." §1.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 76-81, 2003.Glasser, M. L. and Zucker, I. J. "Lattice Sums." In Perspectives in Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring).Zucker, I. J. "Exact Results for Some Lattice Sums in 2, 4, 6 and 8 Dimensions." J. Phys. A: Nucl. Gen. 7, 1568-1575, 1974.Referenced on Wolfram|Alpha
Lattice SumCite this as:
Weisstein, Eric W. "Lattice Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LatticeSum.html