Cubic lattice sums include the following:
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(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where the prime indicates that the origin 
, 
, etc. is excluded from the sum (Borwein and Borwein
1986, p. 288).
These have closed forms for even 
,
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(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
for ![R[s]>1](/images/equations/LatticeSum/Inline25.svg)
, where 
is the Dirichlet
beta function, 
is the Dirichlet eta function, and 
is the Riemann
zeta function (Zucker 1974, Borwein and Borwein 1987, pp. 288-301). The
lattice sums evaluated at 
are called the Madelung constants. An additional
form for 
is given by
 |
(8)
|
for ![R[s]>1/3](/images/equations/LatticeSum/Inline31.svg)
, where 
is the sum of squares
function, i.e., the number of representations of 
by two squares (Borwein and Borwein 1986, p. 291). Borwein
and Borwein (1986) prove that 
converges (the closed form for 
above does not apply for 
), 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),](/images/equations/LatticeSum/NumberedEquation2.svg) |
(9)
|
where 
.
This Madelung constant is expressible in closed
form for 
as
 |
(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),](/images/equations/LatticeSum/NumberedEquation4.svg) |
(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)](/images/equations/LatticeSum/NumberedEquation5.svg) |
(12)
|
(Borwein and Borwein 1986, p. 303), and
 |
(13)
|
(Borwein and Borwein 1986, p. 305).
