A Minimum Property for Cuboidal Lattice Sums
We analyse a family of lattices considered by Conway and Sloane and show that the corresponding Epstein zeta function attains a local minimum for the body-centred cubic lattice.
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| citation_txt | A Minimum Property for Cuboidal Lattice Sums. Shaun Cooper and Peter Schwerdtfeger. SIGMA 21 (2025), 019, 7 pages |
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| description | We analyse a family of lattices considered by Conway and Sloane and show that the corresponding Epstein zeta function attains a local minimum for the body-centred cubic lattice.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 019, 7 pages
A Minimum Property for Cuboidal Lattice Sums
Shaun COOPER a and Peter SCHWERDTFEGER b
a) School of Mathematical and Computational Sciences, Massey University Albany,
Private Bag 102904, North Shore Mail Centre, Auckland 0745, New Zealand
E-mail: s.cooper@massey.ac.nz
b) Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study
(NZIAS), Massey University Albany, Private Bag 102904, North Shore Mail Centre,
Auckland 0745, New Zealand
E-mail: peter.schwerdtfeger@gmail.com
Received January 13, 2025, in final form March 16, 2025; Published online March 24, 2025
https://doi.org/10.3842/SIGMA.2025.019
Abstract. We analyse a family of lattices considered by Conway and Sloane and show that
the corresponding Epstein zeta function attains a local minimum for the body-centred cubic
lattice.
Key words: body-centred cubic; face-centred cubic; Epstein zeta function; lattice sum
2020 Mathematics Subject Classification: 11E45; 11H31
To Stephen Milne on the occasion
of his 75th birthday
1 Introduction
A lattice sum is an expression of the form∑
x
F (x),
where the vector x ranges over a d dimensional lattice. In the case when
F (x) =
1
|x|s
if x ̸= 0,
0 if x = 0,
the lattice sum is a special case of Epstein’s zeta function and has applications in number
theory and theoretical chemistry, e.g., see [2] for comprehensive information or [3] for some
recent applications in chemistry.
Rankin [15] showed that among two-dimensional lattice sums with discriminant 1, the Epstein
zeta function is minimised by the hexagonal lattice. Further results on minimising lattice sums
have been given in [1, 4, 7, 8, 9, 11, 13, 16].
In this work, we consider a one-parameter continuous family of lattices in three dimensions
that includes the face-centred and body-centred cubic lattices, and show that the Epstein zeta
function for the family has a local minimum that is attained by the body-centred cubic lattice.
This paper is a contribution to the Special Issue on Basic Hypergeometric Series Associated with Root
Systems and Applications in honor of Stephen C. Milne’s 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Milne.html
mailto:s.cooper@massey.ac.nz
mailto:peter.schwerdtfeger@gmail.com
https://doi.org/10.3842/SIGMA.2025.019
https://www.emis.de/journals/SIGMA/Milne.html
2 S. Cooper and P. Schwerdtfeger
2 Cuboidal lattices
Following Conway and Sloane [5], assume u and v are positive real numbers, let A = u2/v2, and
consider the cuboidal lattice Λ(u, v) generated by
r1 = (u, v, 0), r2(u, 0, v) and r3 = (0, v, v), (2.1)
that is,
Λ(u, v) = {c1r1 + c2r2 + c3r3 | c1, c2, c3 ∈ Z}. (2.2)
The most important cases, ordered by decreasing value of A, are
(1) A = 1: the face-centred cubic (fcc) lattice,
(2) A = 1/
√
2: the mean centred-cuboidal (mcc) lattice,
(3) A = 1/2: the body-centred cubic (bcc) lattice, and
(4) A = 1/3: the axial centred-cuboidal (acc) lattice.
Thus, Λ(u, v) is a continuous family that includes the fcc and bcc lattices which are well known
to chemistry students. The other two lattices are less well known. The mcc lattice is the unique
densest three-dimensional isodual lattice [5, Theorem 2], while the acc lattice is the least dense
lattice with kissing number1 10, e.g., see [5, 10, 14].
In the remainder of this section, we do some standard calculations to find the minimum norm
and packing density for the lattice Λ(u, v), and use them to define the normalised lattice and
associated Epstein zeta function. If
x = c1r1 + c2r2 + c3r3 ∈ Λ(u, v),
then
∥x∥2 = (c1r1 + c2r2 + c3r3) · (c1r1 + c2r2 + c3r3) = (c1, c2, c3)G(c1, c2, c3)
T,
where G is the Gram matrix whose (i, j) entry is the dot product ri · rj , that is,
G =
u2 + v2 u2 v2
u2 u2 + v2 v2
v2 v2 2v2
= v2
A+ 1 A 1
A A+ 1 1
1 1 2
. (2.3)
The minimum norm of a lattice Λ is [6, p. 42]
µ = min
{
∥x− y∥2 | x,y ∈ Λ, x ̸= y
}
= min
{
∥x∥2 | x ∈ Λ, x ̸= 0
}
.
For the lattice Λ(u, v), we have
µ = min
{
∥x∥2 | x ∈ Λ(u, v), x ̸= 0
}
= min
{
(c1, c2, c3)G(c1, c2, c3)
T | c1, c2, c3 ∈ Z, (c1, c2, c3) ̸= (0, 0, 0)
}
= min
(c1,c2,c3)∈Z3
(c1,c2,c3)̸=0
v2
(
(A+ 1)c21 + (A+ 1)c22 + 2c23 + 2Ac1c2 + 2c1c3 + 2c2c3
)
= min
(c1,c2,c3)∈Z3
(c1,c2,c3)̸=0
v2
(
A(c1 + c2)
2 + (c2 + c3)
2 + (c1 + c3)
2
)
1See [6, p. 21].
A Minimum Property for Cuboidal Lattice Sums 3
and it follows that
µ =
4Av2 if 0 < A < 1/3,
(A+ 1)v2 if 1/3 ≤ A ≤ 1,
2v2 if A > 1.
(2.4)
The packing radius of a lattice is defined by p = 1
2
√
µ and the packing density is [6, pp. 1–7]
∆ =
volume of one sphere of radius p
(detG)1/2
.
From (2.3), we have detG = 4Av6, hence the packing density of Λ(u, v) is given by
∆ =
2πA
3
if 0 < A < 1/3,
π
12
√
(A+ 1)3
A
if 1/3 ≤ A ≤ 1,
π
6
√
2
A if A > 1.
(2.5)
The maximum packing density for this family of lattices is π
√
2
6 ≈ 74% for the fcc lattice at A = 1.
This was conjectured by Kepler and proved by Hales [12] to be maximal among all packings in
three dimensions. Since
d
dA
(
π
12
√
(A+ 1)3
A
)
=
π
24
(
A+ 1
A3
)1/2
(2A− 1),
the packing density has a local minimum at A = 1/2, corresponding to bcc. Some special values
of the density, together with the kissing numbers, are given in Table 1. A graph of the packing
density as a function of A is shown in Figure 1.
Table 1. Packing density and kissing number as functions of A. Here (a, b) means {A | a < A < b}.
A (0, 13)
1
3 (13 ,
1
2)
1
2 (12 ,
1√
2
) 1√
2
( 1√
2
, 1) 1 (1,∞)
name acc bcc mcc fcc
density 2πA
3
2π
9
π
12
√
(A+1)3
A
π
√
3
8
π
12
√
(A+1)3
A
π
√
2
6
π
6
√
2
A
kiss. no. 2 10 8 12 4
In order to study physical properties of the lattice, we normalise so that the minimum distance
is µ = 1. Accordingly, from (2.4) let
v =
1
2
√
A
if 0 < A < 1/3,
1√
A+ 1
if 1/3 ≤ A ≤ 1,
1√
2
if A > 1.
(2.6)
The cases 0 < A < 1/3 and A > 1 are of less interest because the normalised lattice degenerates
to a lattice congruent to Z or Z2 in the limits A → 0 and A → ∞, respectively. From now
4 S. Cooper and P. Schwerdtfeger
A
∆
11
2
1
3
fccbccacc
π
√
2/6 ≈ 0.740
2π/9 ≈ 0.698
π
√
3/8 ≈ 0.680
Figure 1. Graph of ∆ as a function of A given by (2.5).
on, we assume 1/3 ≤ A ≤ 1. From (2.3) and (2.6), the Gram matrix for the normalised
lattice is
G =
1
A+ 1
A+ 1 A 1
A A+ 1 1
1 1 2
.
The associated quadratic form is
g(A; i, j, k) = (i, j, k)G(i, j, k)T
=
1
A+ 1
(
(A+ 1)i2 + (A+ 1)j2 + 2k2 + 2Aij + 2ik + 2jk
)
=
1
A+ 1
(
A(i+ j)2 + (j + k)2 + (i+ k)2
)
and the corresponding Epstein zeta function is
L(A; s) =
∑
i,j,k∈Z
′
(
1
g(A; i, j, k)
)s
=
∑
i,j,k∈Z
′
(
A+ 1
A(i+ j)2 + (j + k)2 + (i+ k)2
)s
, (2.7)
where the primes indicate that the term corresponding to (i, j, k) = (0, 0, 0) is omitted from the
summations. The series (2.7) converges for s > 3/2 and we further assume 1/3 ≤ A ≤ 1.
3 A minimum property
Numerical calculations in [3] suggest that for a fixed value of s > 3/2, the Epstein zeta func-
tion (2.7) appears to have a local minimum value at A = 1/2. This is confirmed by the follow-
ing result.
Theorem 1. Suppose s > 3/2. The Epstein zeta function L(A; s) defined by (2.7) satisfies
∂
∂A
L(A; s)
∣∣∣∣
A=1/2
= 0 and
∂2
∂A2
L(A; s)
∣∣∣∣
A=1/2
> 0.
A Minimum Property for Cuboidal Lattice Sums 5
Proof. By definition, we have
L(A; s) =
∑
I,J,K∈Z
′
(
1
g(A; I, J,K)
)s
,
where
g(A; I, J,K) =
1
A+ 1
(
A(I + J)2 + (J +K)2 + (I +K)2
)
.
Now make the change of variables (I, J,K) = (i − j,−k, j). This is a bijection since (i, j, k) =
(I +K,K,−J), and it follows that
L(A; s) =
∑
i,j,k∈Z
′
(
1
g(A; i− j,−k, j)
)s
=
∑
i,j,k∈Z
′ 1(
i2 + j2 + k2 − 2(ij + ik)
(
A
A+1
)
+ 2jk
(
A−1
A+1
))s .
By direct calculation, the derivative is given by
∂
∂A
L(A; s) =
2s
(A+ 1)2
∑
i,j,k∈Z
′ ij + ik − 2jk(
i2 + j2 + k2 − 2(ij + ik)
(
A
A+1
)
+ 2jk
(
A−1
A+1
))s+1 . (3.1)
Setting A = 1/2 gives
∂
∂A
L(A; s)
∣∣∣∣
A=1/2
=
8s
9
∑
i,j,k∈Z
′ ij + ik − 2jk(
i2 + j2 + k2 − 2
3(ij + ik + jk)
)s+1 . (3.2)
Switching i and j gives
∂
∂A
L(A; s)
∣∣
A=1/2
=
8s
9
∑
i,j,k∈Z
′ ij + jk − 2ik(
i2 + j2 + k2 − 2
3(ij + ik + jk)
)s+1 , (3.3)
while switching i and k in (3.2) gives
∂
∂A
L(A; s)
∣∣∣∣
A=1/2
=
8s
9
∑
i,j,k∈Z
′ jk + ik − 2ij(
i2 + j2 + k2 − 2
3(ij + ik + jk)
)s+1 . (3.4)
On adding (3.2)–(3.4) and noting that
(ij + ik − 2jk) + (ij + jk − 2ik) + (jk + ik − 2ij) = 0,
it follows that
∂
∂A
L(A; s)
∣∣∣∣
A=1/2
= 0.
Next, taking the derivative of (3.1) gives
∂2
∂A2
L(A; s) =
−4s
(A+ 1)3
∑
i,j,k∈Z
′ ij + ik − 2jk(
i2 + j2 + k2 − 2(ij + ik)
(
A
A+1
)
+ 2jk
(
A−1
A+1
))s+1
+
4s(s+ 1)
(A+ 1)4
∑
i,j,k∈Z
′ (ij + ik − 2jk)2(
i2 + j2 + k2 − 2(ij + ik)
(
A
A+1
)
+ 2jk
(
A−1
A+1
))s+2 .
6 S. Cooper and P. Schwerdtfeger
0.4 0.6 0.8 1
8
10
12
14
Figure 2. Graphs of y = L(A; s) for 1/3 ≤ A ≤ 1 given by (2.7) for (from top to bottom) s = 3, s = 6,
s = 20 and s = ∞. In the limiting case s → ∞ we have L(A;∞) = kiss(A) where kiss(A) is the kissing
number as a function of A as given in Table 1.
When A = 1/2 the first sum is zero by the calculations in the first part of the proof. Therefore,
∂2
∂A2
L(A; s)
∣∣∣∣
A=1/2
=
64s(s+ 1)
81
∑
i,j,k∈Z
′ (jk + ik − 2ij)2(
i2 + j2 + k2 − 2
3(ij + ik + jk)
)s+2 .
The term (jk+ik−2ij)2 in the numerator is non-negative and not always zero. The denominator
is always positive because the quadratic form is positive definite. It follows that
∂2
∂A2
L(A; s)
∣∣∣∣
A=1/2
> 0
as required.
The calculations above are valid provided term-by-term differentiation of the series is allowed.
All of the series encountered above converge absolutely and uniformly on compact subsets of
the region Re(s) > 3/2. On restricting s to real values, the conclusion about positivity is valid
for s > 3/2. ■
A consequence of Theorem 1 is that for any fixed value s > 3/2, the lattice sum L(A; s)
attains a local minimum when A = 1/2. The values s = 6 and s = 3 are used in the classical
Lennard–Jones 12-6 potential, so the values s > 3/2 are sufficient for physical applications.
Some graphs of y = L(A; s) for 1/3 ≤ A ≤ 1 for various values of s are given in Figure 2.
A The limiting cases A → 0 and A → ∞
In Section 2, it was stated that the normalised lattices are congruent to Z or Z2 in the lim-
its A → 0 and A → ∞, respectively. Here we provide an analysis for the case A → ∞. When
A > 1, by (2.1), (2.2) and (2.6) the normalised lattice is
Λ =
{
c1
(√
A
2
,
1√
2
, 0
)
+ c2
(√
A
2
, 0,
1√
2
)
+ c3
(
0,
1√
2
,
1√
2
)
| c1, c2, c3 ∈ Z
}
.
In the limit A → ∞, the only vectors that remain finite are those with c2 = −c1, hence the
limiting set is given by
Λ∞ := lim
A→∞
Λ =
{
c1
(
0,
1√
2
,
−1√
2
)
+ c3
(
0,
1√
2
,
1√
2
)
| c1, c3 ∈ Z
}
.
A Minimum Property for Cuboidal Lattice Sums 7
The function ϕ : Λ∞ → Z2 given by
ϕ
(
c1
(
0,
1√
2
,
−1√
2
)
+ c3
(
0,
1√
2
,
1√
2
))
= (c1, c3)
is an isometry, i.e., it preserves distance, hence the normalised lattice in the limiting case A → ∞
is congruent to Z2. The other limiting case A → 0 may be analysed in a similar way by
considering the normalised lattice in the case 0 < A < 1/3. We omit the details as they are
similar.
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1 Introduction
2 Cuboidal lattices
3 A minimum property
A The limiting cases A -> 0 and A -> infty
References
|
| id | nasplib_isofts_kiev_ua-123456789-212872 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:30:48Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cooper, Shaun Schwerdtfeger, Peter 2026-02-13T13:48:58Z 2025 A Minimum Property for Cuboidal Lattice Sums. Shaun Cooper and Peter Schwerdtfeger. SIGMA 21 (2025), 019, 7 pages 1815-0659 2020 Mathematics Subject Classification: 11E45; 11H31 arXiv:2501.05746 https://nasplib.isofts.kiev.ua/handle/123456789/212872 https://doi.org/10.3842/SIGMA.2025.019 We analyse a family of lattices considered by Conway and Sloane and show that the corresponding Epstein zeta function attains a local minimum for the body-centred cubic lattice. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Minimum Property for Cuboidal Lattice Sums Article published earlier |
| spellingShingle | A Minimum Property for Cuboidal Lattice Sums Cooper, Shaun Schwerdtfeger, Peter |
| title | A Minimum Property for Cuboidal Lattice Sums |
| title_full | A Minimum Property for Cuboidal Lattice Sums |
| title_fullStr | A Minimum Property for Cuboidal Lattice Sums |
| title_full_unstemmed | A Minimum Property for Cuboidal Lattice Sums |
| title_short | A Minimum Property for Cuboidal Lattice Sums |
| title_sort | minimum property for cuboidal lattice sums |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212872 |
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