TOPICS
Search

Lattice Sum


Cubic lattice sums include the following:

b_2(2s)=sum^'_(i,j=-infty)^infty((-1)^(i+j))/((i^2+j^2)^s)
(1)
b_3(2s)=sum^'_(i,j,k=-infty)^infty((-1)^(i+j+k))/((i^2+j^2+k^2)^s)
(2)
b_n(2s)=sum^'_(k_1,...,k_n=-infty)^infty((-1)^(k_1+...+k_n))/((k_1^2+...+k_n^2)^s),
(3)

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,

b_2(2s)=-4beta(s)eta(s)
(4)
b_4(2s)=-8eta(s)eta(s-1)
(5)
b_6(2s)=4beta(s-2)eta(s)-16beta(s)eta(s-2)
(6)
b_8(2s)=-16zeta(s)eta(s-3)
(7)

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

 b_2(2s)=sum_(n=1)^infty((-1)^nr_2(n))/(n^s)
(8)

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

 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)

giving the special case

 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)
(12)

(Borwein and Borwein 1986, p. 303), and

 sum^'_(k,m,n=-infty)^infty((-1)^(k+m+n+1))/((|k|+|m|+|n)^s)=2eta(s)+4eta(s-2)
(13)

(Borwein and Borwein 1986, p. 305).


See also

Benson's Formula, Double Series, Epstein Zeta Function, Grenz-Formel, Madelung Constants

Explore with Wolfram|Alpha

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 Sum

Cite this as:

Weisstein, Eric W. "Lattice Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LatticeSum.html

Subject classifications