application of an idea of voronoĭ to lattice zeta functions

ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2012, Vol. 276, pp. 103–124. c© Pleiades Publishing, Ltd., 2012.

Application of an Idea of Voronoıto Lattice Zeta Functions

Peter M. Gruber a

Received July 2011

In memoriam Anatoliı A. Karatsuba (1937–2008)

Abstract—A major problem in the geometry of numbers is the investigation of the local min-ima of the Epstein zeta function. In this article refined minimum properties of the Epstein zetafunction and more general lattice zeta functions are studied. Using an idea of Voronoı, charac-terizations and sufficient conditions are given for lattices at which the Epstein zeta function isstationary or quadratic minimum. Similar problems of a duality character are investigated forthe product of the Epstein zeta function of a lattice and the Epstein zeta function of the polarlattice. Besides Voronoı type notions such as versions of perfection and eutaxy, these resultsinvolve spherical designs and automorphism groups of lattices. Several results are extended tomore general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.

DOI: 10.1134/S0081543812010099

1. INTRODUCTION

The aim of the article is to investigate (local) minimum properties of the Epstein zeta functionand of a generalization of it, using an idea of Voronoı.

The key idea behind Voronoı’s [43–45] work in the geometric theory of positive definite quadraticforms on E

d can be described as follows: If positive definite quadratic forms are identified with theircoefficient vectors, certain problems on forms are translated into geometric problems in E

12d(d+1)

which, sometimes, are easier to deal with. In this way Voronoı proved the famous criterion: a positivedefinite quadratic form is extreme if and only if it is perfect and eutactic. Earlier similar ideas inthe context of projective geometry are due to Monge, Plucker and Klein.

While largely ignored in the first decades after its publication, the idea of Voronoı has seennumerous applications since the 1960s. Voronoı type investigations of the Epstein zeta function goback to Delone, Ryshkov [13, 33], their students and the British school of the geometry of numbers,including Rankin [32], Cassels [9] and Montgomery [31]. See also Ennola [15]. Recent contributionsare due to Sarnak and Strombergsson [34] and Coulangeon [11]. Ash [1] showed that the density oflattice packings of balls is a Morse function. Berge and Martinet [7] studied duality problems for thedensity. Their investigations were continued by the author [25]. Lattice coverings with balls wereinvestigated by Barnes and Dickson [4], Delone et al. [12], Schurmann and Vallentin [41] and theauthor [21]. For uniqueness problems of lattice packings and coverings of extreme density, see theauthor [23]. The kissing number of lattice packings is related to the number of closed geodesics onthe Riemannian manifolds of a Teichmuller space (see Bavard [5, 6] and Schmutz Schaller [35–38]).The author [20] applied the idea of Voronoı to John type and minimum position problems in theasymptotic theory of normed spaces. See also the articles of the author and Schuster [28] and theauthor [18, 22]. Surveys are [26, 24].

a Institut fur Diskrete Mathematik und Geometrie, Technische Universitat Wien, Wiedner Hauptstraße 8-10/104,A-1040 Vienna, Austria.E-mail address: [email protected]

103

104 P.M. GRUBER

The Epstein zeta function is defined by

ζ = ζ(L, s) =∑

l∈L\{o}

1‖l‖s

for s > d,

where L is a lattice in Ed of determinant 1 and ‖ · ‖ the usual Euclidean norm. The function ζ is

important for physics (potentials of crystals, field theory, and Casimir effect), numerical integrationand other fields (see the remarks and references in the articles of Sarnak and Strombergsson [34]and Lim and Teo [29]). In the context of physics and the geometry of numbers a major problemfor ζ is the investigation of local minimum properties of ζ as L ranges over the space of all latticesin E

d of determinant 1 for a given s > d, for all sufficiently large s, or for all s > d. A problemrelevant to physics is the investigation of the zeta function ζ∗ of the polar lattice L∗.

This article contains characterizations and sufficient conditions for lattices at which ζ and ζζ∗ are

stationary andquadratic minimum,

for certain (possibly all) s > d. In dimensions 2 and 3 the lattices which provide minima of ζ and ζζ∗

are specified. A relation between minimum properties of the Epstein zeta function and maximumproperties of the density of lattice packings of balls is stated. Most of these results extend to moregeneral lattice zeta functions ζC , where the Euclidean norm is replaced by an arbitrary smooth norm.

Our characterizations and sufficient conditions make use of versions of

perfection,eutaxy, and ofspherical designs andautomorphism groups of lattices.

It is surprising to see how well these notions are suited to characterize stationarity and quadraticminimality of lattice zeta functions. We also observed the same phenomenon in the study of localextremum properties of the density of lattice packings and coverings of balls (see [25, 21]). Moreover,eutaxy and dual eutaxy, properly generalized, are perfect tools for John’s ellipsoid theorem and itsgeneralizations, for the Loewner ellipsoid, and for the characterization of minimum ellipsoidal shells(cf. the author and Schuster [28] and the author [20, 22]).

Several open problems are collected in the last section.For information on convex geometry and the geometry of numbers we refer to [39, 27, 19]. For

positive definite quadratic forms, their geometric theory, spherical designs, automorphism groups,and lattice packing of balls consult [10, 46, 30, 19, 40].

Let the symbols tr, lin, #, ‖ · ‖, · , V , Bd, Sd−1, and T stand for trace, linear hull, cardinalnumber, Euclidean norm, inner product, volume, unit ball and unit sphere in Euclidean d-space E

d,and transposition.

2. MINIMUM PROPERTIES OF ζ

A result of Delone and Ryshkov [13] says that ζ is minimum at a lattice L for all sufficientlylarge s if and only if L is perfect and each layer of L is eutactic or, more precisely and in a differentterminology, is a spherical 2-design. If each layer of L is a spherical 4-design, then ζ is minimumfor all s > d as shown by Coulangeon [11]. A different sufficient minimum condition with numerousapplications is due to Sarnak and Strombergsson [34]. These authors show that ζ is minimum formany lattices, including the lattices which provide the densest lattice packings of balls in dimensionsd = 2, 3, . . . , 8, the Barnes–Wall and the Leech lattice.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 276 2012

APPLICATION OF AN IDEA OF VORONOI 105

In this section, refined minimum properties of the Epstein zeta function are studied. First, thelattices are characterized at which ζ is stationary for a fixed s > d and for all s > d (Theorem 1and Corollary 1). The results involve versions of the notion of eutaxy. A sufficient condition forstationarity for each s > d follows from Corollary 1 and Proposition 2. It involves the automorphismgroup of the lattice. Next, lattices which provide quadratic minima of ζ for a fixed s > d and forall sufficiently large s are characterized. Sufficient conditions for lattices which provide quadraticminima of ζ for each s > d are specified (Theorems 2 and 3 and Corollary 4). The conditions makeuse of Voronoı type variants of the concept of eutactic lattice, and of automorphism groups. Theseresults are used in dimensions 2 and 3 to specify the lattices for which ζ is stationary or quadraticminimum (Corollaries 2 and 5). It turns out that the lattices at which ζ is quadratic minimumfor arbitrarily large s provide lattice packings of balls of ultra-maximum density, as studied in [25](Corollary 6).

Some of the conditions which characterize lattices with refined extremum properties are difficultto check. Therefore, we have stated several sufficient conditions which are easier to verify and stillapply in many cases (Corollaries 3 and 4).

Basic notions. A lattice L in Euclidean d-space Ed is the set of all integer linear combinations

of d linearly independent vectors in Ed. The volume of the parallelotope spanned by these vectors

is the determinant d(L) of L. Let L be a lattice with d(L) = 1.Real symmetric d × d matrices are identified with points in E

12d(d+1). The symbols O and I

stand for the d × d zero and unit matrix, respectively. For A = (aik), B = (bik) ∈ E12d(d+1) the

inner product A ·B and the norm ‖A‖ are defined to be∑

aikbik and(∑

a2ik

)1/2, respectively. Thedot · and the norm ‖ · ‖ also denote the usual inner product and the corresponding Euclidean normin E

d. Let T be the subspace

T ={A ∈ E

12d(d+1) : tr A = A · I = 0

}

of E12d(d+1) of codimension 1. For l, u ∈ E

d their tensor product l⊗u is the d×d matrix luT = (liuk).

Minimum and Voronoı type properties. Let s > d. Then ζ(·, s) is stationary, minimum,or quadratic minimum at the lattice L for s if

ζ(

I+Adet(I+A)1/d L, s

)

ζ(L, s)

⎧⎪⎨

⎪⎩

= 1 + o(‖A‖),≥ 1,

≥ 1 + const · s2‖A‖2

as A → O, A ∈ T .

We say that ζ is stationary, etc., for a range of s if it is stationary, etc., for each individual s. If, inthe following, the symbols const > 0, o(·) and O(·), which depend only on L, appear several timesin the same context, they may be different each time. An inequality holds as A → O or t → 0 if itholds for all A, respectively t, for which ‖A‖, respectively |t|, is sufficiently small. Note that all theabove minimum properties are local.

For n = 1, 2, . . . , a spherical n-design is a subset M = {±l1, . . . ,±lk} of the unit sphere Sd−1,such that for any polynomial p : E

d → R of degree at most n, the following equality holds:∫

Sd−1

p(u) dσ(u) =12k

∑

l∈M

p(l),

where σ is the usual rotation invariant area measure on Sd−1, normalized so that Sd−1 has area 1.Venkov [42] showed that

a finite o-symmetric set M ⊆ Sd−1 is a spherical n-design

if and only if∑

l∈M (l · x)n = const · ‖x‖n for x ∈ Ed.

(1)

The points of L \ {o} with the same norm form a layer of L.


106 P.M. GRUBER

We are now in a position to state the following definitions.Definition 1. A layer M of L is strongly eutactic, or (after a suitable normalization) is a

spherical 2-design, if it satisfies one of the following three equivalent conditions, where 2k = #M :

∑

l∈M

l ⊗ l

‖l‖2=

2kd

I,∑

l∈M

A · l ⊗ l

‖l‖2= 0 for A ∈ T ,

∑

l∈M

(l · x)2

‖l‖2=

2kd‖x‖2 for x ∈ E

d.

This notion of strong eutaxy is a special case of the classical concept of eutaxy, where thecoefficients of the tensor products l ⊗ l are all positive, but may be different. The lattice L isstrongly eutactic if its first layer is.

Definition 2. The lattice L is fully eutactic for s if it satisfies one of the following threeequivalent conditions:

∑

l∈L\{o}

l ⊗ l

‖l‖s+2=

ζ(L, s)d

I,∑

l∈L\{o}

A · l ⊗ l

‖l‖s+2= 0 for A ∈ T ,

∑

l∈L\{o}

(l · x)2

‖l‖s+2=

ζ(L, s)d

‖x‖2 for x ∈ Ed.

The equivalence of the conditions in Definition 1, as well as in Definition 2, can be verified easily.In view of the results on spherical designs by Goethals and Seidel [17], Bachoc and Venkov [3] andothers, proposition (1) shows that many special lattices in the literature are strongly eutactic, orall their layers (with obvious normalization) are strongly eutactic.

Refined versions of strong eutaxy and full eutaxy are as follows. The equivalence of the twoconditions in the first definition is due to Venkov, and the proof for the second definition followsthe same pattern.

Definition 3. A layer M of L is ultra eutactic, or is a spherical 4-design, if it has one of thefollowing equivalent properties, where 2k = #M :

∑

l∈M

(A · l ⊗ l)2

‖l‖4=

4kd(d + 2)

‖A‖2 +2k

d(d + 2)(tr A)2 for A ∈ E

12d(d+1),

∑

l∈M

(l · x)4

‖l‖4=

6kd(d + 2)


Definition 4. The lattice L is completely eutactic for s if it has one of the following equivalentproperties:

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4=

2ζ(L, s)d(d + 2)

‖A‖2 +ζ(L, s)d(d + 2)

(tr A)2 for A ∈ E12d(d+1),

∑

l∈L\{o}

(l · x)4

‖l‖s+4=

3ζ(L, s)d(d + 2)


A layer M of L is perfect if

E12d(d+1) = lin{l ⊗ l : l ∈ M}.

If M is the first layer, then L is called perfect. The automorphism (or symmetry) group A = A(L)of L consists of all orthogonal transformations of E

d which fix the origin o and map L onto itself.



Characterization of stationarity.Theorem 1. Let s > d. Then the following properties of ζ and L are equivalent :

(i) ζ is stationary at L for s;

(ii) L is fully eutactic for s.

There are numerous stationary lattices, for example, the lattices for which ζ(·, s) attains itsabsolute minimum. The next result contains an effective tool to determine lattices which arestationary for each s > d.

Corollary 1. The following properties of ζ and L are equivalent :

(i) ζ is stationary at L for each s > d;

(ii) ζ is stationary at L for arbitrarily large s;(iii) each layer of L is strongly eutactic;(iv) L is fully eutactic for each s > d.

Stationarity in dimensions 2, 3 and in general dimensions. Berge and Martinet [8]and Bavard [6] gave descriptions of the lattices in dimensions 2, 3, and 4 which are eutactic in theusual sense. Ash [2] and these authors showed that in general dimensions there are only finitelymany similarity classes of eutactic lattices; for an alternative proof see [26]. Scrutiny of the eutacticlattices in dimensions 2 and 3 together with the latter result yields the following remark.

Corollary 2. (i) In dimensions 2 and 3 the zeta function ζ is stationary for all s > d preciselyat the following lattices of determinant 1:

d = 2: tp square, hp hexagonal ;d = 3: cP primitive cubic, cI body centered cubic, cF face centered cubic.(ii) In general dimensions there are, up to orthogonal transformations, only finitely many lattices

of determinant 1 at which ζ is stationary for all s > d.Up to dilatations, these lattices in E

2 and E3 constitute Bravais classes. Bravais classes form a

classification of lattices in crystallography by means of their automorphism groups. Usual symbolsare tp, hp, . . . , cF (cf. Erdos, Gruber and Hammer [16] and Engel [14]).

Characterization and sufficient condition for quadratic minimality. We begin witha necessary and sufficient condition for given s > d, then consider (sets of) arbitrarily large andsufficiently large s and, finally, all s > d.

Theorem 2. Let s > d. Then the following properties of ζ and L are equivalent :

(i) ζ is quadratic minimum at L for s;

(ii) L is fully eutactic for s and satisfies the inequality

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4>

2ζ(L, s)d(s + 2)

‖A‖2 for A ∈ T \ {O}.

Corollary 3. Let s > d. If L is completely eutactic for s, then ζ is quadratic minimum at Lfor s.

Theorem 3. The following properties of ζ and L are equivalent :

(i) ζ is quadratic minimum at L for all sufficiently large s;

(ii) ζ is minimum at L for arbitrarily large s;(iii) L is perfect and each layer of L is strongly eutactic.


108 P.M. GRUBER

The result of Delone and Ryshkov [13] says that the statement that ζ is minimum at L for allsufficiently large s is equivalent to (iii).

Corollary 4. Each of the following conditions (i) and (ii) for L is sufficient for ζ to bequadratic minimum at L for all sufficiently large s:

(i) L is perfect and A(L) is transitive on the first layer M of L;(ii) L is perfect and fully eutactic for each s > d.

Each of the following conditions (iii) and (iv) for L is sufficient for ζ to be quadratic minimumat L for all s > d:

(iii) each layer of L is ultra eutactic;(iv) L is completely eutactic for each s > d.

In any dimension there are lattices such that each layer is ultra eutactic and thus, a fortiori,strongly eutactic. Examples are irreducible root lattices. Coulangeon [11] proved that condition (iii)implies that ζ is minimum at L for all s > d.

Remark. Each of conditions (iii) and (iv) in Corollary 4 implies that L is fully eutactic and

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4=

2ζ(L, s)d(d + 2)

‖A‖2 >2ζ(L, s)d(s + 2)

‖A‖2 for A ∈ T , s > d.

Thus, by Theorem 2, conditions (iii) and (iv) are, by far, sufficient for ζ to be quadratic minimumat L for all s > d. Hence, there seems to be a certain range for sufficient conditions which areweaker than (iii) or (iv), but still guarantee quadratic minimality for each s > d.

Quadratic minimality for d = 2, 3 and general d.Corollary 5. (i) In dimensions 2 and 3 it is precisely the following lattices of determinant 1

at which ζ is quadratic minimum for each s > d, as well as at which ζ is quadratic minimum forall sufficiently large s:

d = 2: hp hexagonal,d = 3: cF face centered cubic.(ii) In general dimensions there are, up to orthogonal transformations, only finitely many lattices

of determinant 1 at which ζ is quadratic minimum for each s > d or for all sufficiently large s.

Relations to lattice packing of balls. The packing radius of the Euclidean unit ball Bd

with respect to L is given by

� = �(Bd, L) = max{σ > 0: {σBd + l : l ∈ L} is a lattice packing

},

where {σBd + l : l ∈ L} is a lattice packing of Bd with packing lattice L if no two of the balls σBd + loverlap. In particular, {�Bd + l : l ∈ L} is a lattice packing. Its density is defined by

δ = δ(Bd, L) =�dV (Bd)

d(L).

The study of the local maximum properties of positive definite quadratic forms is equivalent to thestudy of the local maximum properties of δ(Bd, ·) on the space of lattices. Call δ semi-stationary,maximum, or ultra-maximum at L if

δ(Bd, (I + A)L)δ(Bd, L)

≤

⎧⎪⎨

⎪⎩

1 + o(‖A‖),1,

1 − const · ‖A‖as A → O, A ∈ T .



In [25] it was proved that δ(Bd, ·) is not stationary at any lattice and that it is semi-stationaryor ultra-maximum if and only if L is semi-eutactic or, respectively, perfect and eutactic in the usualsense. (Thus, in particular, maximality equals ultra-maximality.) These results and Theorems 1and 3 together yield the following statements.

Corollary 6. Among the extremum properties of ζ and δ the following relations hold :(i) If ζ is stationary at L for arbitrarily large s, then δ is semi-stationary at L.(ii) If ζ is quadratic minimum at L for arbitrarily large s, then δ is ultra-maximum at L.

Tools for the proofs. To make the proofs of the above results more transparent and compact,we collect technical tools together into a preliminary section for later reference. Simple calculationsyield the identities

‖l ⊗ l‖ = ‖l‖2, l ⊗ l · m ⊗ m = (l · m)2, I · l ⊗ l = l2 = ‖l‖2,

A · l ⊗ l = Al · l = lTAl, (Al)2 = A2 · l ⊗ l,

tr AB = I · AB = A · B, tr A2 = I · A2 = ‖A‖2 for l,m ∈ Ed, A,B ∈ E

12d(d+1);

(2)

det(I + A) = 1 + tr A +12((tr A)2 − A · AT

)+ O(‖A‖3) as A → O, A ∈ E

d2,

det(I + A) = 1 − 12A · AT + O(‖A‖3) as A → O, A ∈ S,

det(I + A) = 1 − 12‖A‖2 + O(‖A‖3) as A → O, A ∈ T ,

(3)

where d × d matrices which are not necessarily symmetric are identified with points in Ed2 and

S = {A ∈ Ed2

: A · I = tr A = 0}. To see this, note that

det(I + A) =

∣∣∣∣∣∣∣∣∣∣

1 + a11 a12 a13 . . . a1d

a21 1 + a22 a23 . . . a2d

a31 a32 1 + a33 . . . a3d

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ad1 ad2 ad3 . . . 1 + add

∣∣∣∣∣∣∣∣∣∣

= 1 + tr A +∑

i<j

aiiajj −∑

i<j

aijaji + O(‖A‖3)

= 1 + tr A +12

(∑

i

aii

)2

− 12

∑

i

a2ii −

12

∑

i�=j

aijaji + O(‖A‖3)

= 1 + tr A +12((tr A)2 − A · AT

)+ O(‖A‖3) as A → O, A ∈ E

d2.

The equality

(l + Al)2 = l2(1 + 2A · n ⊗ n + A2 · n ⊗ n) for A ∈ E12d(d+1), l ∈ E

d, n =l

‖l‖ ,

implies the formula1

‖l + Al‖s=

1‖l‖s

(1 + 2A · n ⊗ n + A2 · n ⊗ n

)−s/2

=1

‖l‖s

(1 − sA · n ⊗ n − s

2A2 · n ⊗ n +

s(s + 2)2

(A · n ⊗ n)2 + O(‖A‖3))

=1

‖l‖s

(1 − sA · n ⊗ n + O(‖A‖2)

)as A → O, A ∈ T , l ∈ E

d \ {o}, n =l

‖l‖ , (4)

where O(·) may be chosen to be the same for each l ∈ L \ {o}.


110 P.M. GRUBER

In the following, if not indicated otherwise, ζ stands for ζ(L, s). The definition of ζ and identi-ties (3) and (4) yield the representation

ζ

(I + A

det(I + A)1/dL, s

)= ζ

((I + A)L, s

)det(I + A)s/d

=∑

l∈L\{o}

1‖l‖s

(1 − sA · n ⊗ n − s

2A2 · n ⊗ n +

s(s + 2)2

(A · n ⊗ n)2 + O(‖A‖3))

×(

1 − 12‖A‖2 + O(‖A‖3)

)s/d

= ζ − sA ·∑

l∈L\{o}

n ⊗ n

‖l‖s− s

2A2 ·

∑

l∈L\{o}

n ⊗ n

‖l‖s+

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s

− sζ

2d‖A‖2 + ζO(‖A‖3) as A → O, A ∈ T . (5)

If L is fully eutactic, this is equal to

ζ − 0 − sζ

d‖A‖2 +

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s+ ζO(‖A‖3)

as A → O, A ∈ T , for arbitrarily large s.

Let Li = {±li1, . . . ,±liki}, i = 1, 2, . . . , be the layers of L.

Proposition 1. The lattice L has the following properties :(i) If each layer of L is strongly eutactic or ultra eutactic, then L is fully eutactic or completely

eutactic, respectively, for each s > d.(ii) If L is fully eutactic or completely eutactic for arbitrarily large s, then each layer of L is

strongly eutactic or ultra eutactic, respectively.(iii) If L is completely eutactic for an s > d, then it is fully eutactic for s.

Proof. Since the proofs of statements (i) and (ii) for ultra eutaxy are very similar to the proofsin the case of strong eutaxy, only the proofs for the latter will be presented.

(i) By assumption,∑

l∈Li

l ⊗ l

‖l‖2=

2ki

dI for i = 1, 2, . . . .

Thus∑

l∈L\{o}

l ⊗ l

‖l‖s+2=

1d

∞∑

i=1

2ki

‖li1‖sI =

1d

∑

l∈L\{o}

1‖l‖s

I =ζ

dI for s > d,

concluding the proof that L is fully eutactic for any s > d.(ii) The following equality holds by assumption:

∑

l∈L\{o}

l ⊗ l

‖l‖s+2=

ζ

dI for arbitrarily large s.

Putting

Ni =∑

l∈Li

n ⊗ n for i = 1, 2, . . . ,



we can write this equality as follows:

N1

‖l11‖s+

N2

‖l21‖s+ . . . =

1d

(2k1

‖l11‖s+

2k2

‖l21‖s+ . . .

)I,

i.e.,

N1 +‖l11‖s

‖l21‖sN2 + . . . =

2k1

dI +

‖l11‖s

‖l21‖s

2k2

dI + . . . for arbitrarily large s. (6)

Now, let s tend to +∞. Then

N1 =2k1

dI;

that is, L1 is strongly eutactic. Next, we cancel the first summand on each side of (6) and proceedwith N2 as before with N1. This gives

N2 =2k2

dI;

that is, L2 is strongly eutactic. Continuing in this way, we obtain the strong eutaxy of eachlayer of L.

(iii) By Definition 4, the assumption may be expressed in the form∑

l∈L\{o}

(l · x)4

‖l‖s+4=

3ζd(d + 2)

‖x‖4 for x ∈ Ed and given s > d.

Consider the second derivative with respect to x of both sides of this equality and equate the traces:∑

l∈L\{o}

4(l · x)3l‖l‖s+4

=3ζ

d(d + 2)· 4‖x‖2x =

3ζd(d + 2)

· 4x2x,

∑

l∈L\{o}

3(l · x)2l ⊗ l

‖l‖s+4=

3ζd(d + 2)

(2x ⊗ x + x2I

),

∑

l∈L\{o}

(l · x)2

‖l‖s+2=

ζ

d(d + 2)(2‖x‖2 + d‖x‖2

)=

ζ

d‖x‖2 for x ∈ E

d.

Thus, by Definition 2, L is fully eutactic for s. �Proposition 2. Let M be a layer of L. Then the following properties hold :(i) If M is strongly eutactic and satisfies the inequality

∑

l∈M

(A · l ⊗ l)2 > 0 for A ∈ T \ {O},

then M is perfect.(ii) If M is ultra eutactic, then it is strongly eutactic and perfect.(iii) If M is ultra eutactic and the automorphism group A of L operates transitively on M, then

M is perfect and each layer of L is strongly eutactic.(iv) If M is perfect and A is transitive on M, then each layer of L is strongly eutactic.

Berge and Martinet [7] proved results which are similar to properties (iii) and (iv) in thisproposition.

Proof. (i) We have to show the following:

Let A + αI be an arbitrary vector in E12d(d+1) \ {O} where A ∈ T

and α ∈ R. Then (A + αI) · l ⊗ l = 0 for a suitable l ∈ M .(7)


112 P.M. GRUBER

If α = 0, then A = O and the inequality in (i) shows that A · l ⊗ l = 0 for a suitable l ∈ M . IfA = O, then α = 0 and we have αI · l ⊗ l = α‖l‖2 = 0 for each l ∈ M . Assume now that A = Oand α = 0. If (A + αI) · l ⊗ l = 0 for each l ∈ M , then A · l ⊗ l = −αI · l ⊗ l = −α‖l‖2 and thus,for 2k = #M ,

0 = −2kα = A ·∑

l∈M

l ⊗ l

‖l‖2= A · 2k

dI =

2kd

tr A = 0

by the strong eutaxy of M and since A ∈ T . A contradiction, concluding the proof of (7). Thus,M is perfect.

(ii) (cf. the proof of Proposition 1(iii)) By Definition 3, the assumption in (ii) may be writtenin the form

∑

l∈M

(l · x)4

‖l‖4=

6kd(d + 2)


Equating the traces of the second derivatives of the two sides of this identity shows that M isstrongly eutactic. If M is not perfect, then there is an A ∈ E

12d(d+1) \ {O} such that A · l ⊗ l = 0

for each l ∈ M . Then, by Definition 3, M is not ultra eutactic, a contradiction.(iii) By (ii), M is perfect and strongly eutactic. It remains to show that each layer of L is

strongly eutactic. For this it is sufficient to show the following proposition:

Assume that the layer M is perfect and the automorphism group A of L

acts transitively on M . Then each layer Li of L is strongly eutactic.(8)

For the proof of (8), the following will be shown first:

I · m ⊗ m = const for m ∈ M,∑

l∈Li

l ⊗ l · m ⊗ m = const for m ∈ M, i = 1, 2, . . . . (9)

The first equality is obvious. To see the second one, let m,n ∈ M . By the transitivity of A on M ,there is an automorphism S ∈ A such that m = Sn. Note (2) and the fact that ST = S−1 maps Li

onto itself to see that∑

l∈Li

l ⊗ l · m ⊗ m =∑

l∈Li

(l · m)2 =∑

l∈Li

(l · Sn)2 =∑

l∈Li

(STl · n)2

=∑

l∈STLi

(l · n)2 =∑

l∈Li

(l · n)2 =∑

l∈Li

l ⊗ l · n ⊗ n.

This concludes the proof of (9). Since M is perfect, (9) implies that∑

l∈Li

l ⊗ l = μiI with suitable μi > 0, for i = 1, 2, . . . .

Equating the traces of both sides gives 2ki‖li1‖2 = μid, or μi = 2ki‖li1‖2/d, concluding the proofof the fact that Li is strongly eutactic and thus of (8). The proof of (iii) is complete.

(iv) This is an immediate consequence of (8). �Proofs of the theorems and corollaries.Proof of Theorem 1. Clearly, (5) implies the identity

ζ

(I + A

det(I + A)1/dL, s

)= ζ − sA ·

∑

l∈L\{o}

l ⊗ l

‖l‖s+2+ O(‖A‖2) as A → O, A ∈ T .



With it, the proof of Theorem 1 is simple:

L is stationary ⇔ ζ − sA ·∑

l∈L\{o}

l ⊗ l

‖l‖s+2+ O(‖A‖2) = ζ

(1 + o(‖A‖)

)as A → O, A ∈ T

⇔ A ·∑

l∈L\{o}

l ⊗ l

‖l‖s+2= 0 for A ∈ T =

{A ∈ E

12d(d+1) : A · I = 0

}

⇔∑

l∈L\{o}

l ⊗ l

‖l‖s+2= λI for suitable λ ∈ R

⇔ L is fully eutactic,

where the value ζ/d for λ is obtained by equating the traces of the two sides of the last equality. �Proof of Corollary 1. Theorem 1 and Proposition 1 readily yield Corollary 1. �Proof of Corollary 2. (i) A look into [8] or [6] shows that the strongly eutactic lattices L

with d(L) = 1 in E2 and E

3 are precisely the lattices L with d(L) = 1 which are contained in theBravais classes tp, hp in E

2 and cP, cI, cF in E3. Now apply Theorem 1.

(ii) In [2, 8] it was shown that, up to orthogonal transformations, there are only finitely manyeutactic lattices L in E

d with d(L) = 1. Now apply Theorem 1. �Proof of Theorem 2. (i) ⇔ (ii). From (5) we take the equality

ζ

(I + A

det(I + A)1/dL, s

)= ζ − sζ

d‖A‖2 +

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s+ O(‖A‖3)

as A → O, A ∈ T , if L is fully eutactic for s.

Then the following holds:

L is quadratic minimum and, thus, stationary for s

⇔ ζ

(I + A

det(I + A)1/dL, s

)≥ ζ(1 + const · ‖A‖2) as A → O, A ∈ T ,

and L is fully eutactic for s

⇔ −sζ

d‖A‖2 +

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s≥ ζ · const · ‖A‖2 as A → O, A ∈ T ,

and L is fully eutactic for s

⇔∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+2>

2ζd(s + 2)

‖A‖2 for A ∈ T \ {O} and L is fully eutactic.

Here we have used Theorem 1 and the above equality. In the last equivalence the implication ⇒ isclear. To see the reverse implication ⇐, note that the expression

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4− 2ζ

d(s + 2)‖A‖2

may be considered to be a quadratic form in the variable A ∈ T . It is, clearly, positive definite.Thus it is bounded below by an expression of the form const · ‖A‖2. �


114 P.M. GRUBER

Proof of Corollary 3. By Proposition 1(iii), complete eutaxy for s implies that L is fullyeutactic for s, and Definition 4 of complete eutaxy for s shows that

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4=

2ζd(d + 2)

‖A‖2 >2ζ

d(s + 2)‖A‖2 for A ∈ T \ {O}.

Now apply Theorem 2 to see that L is quadratic minimum for s. �Proof of Theorem 3. (i) ⇒ (ii). Obvious.(ii) ⇒ (iii). By (5) we have the identity

ζ

(I + A

det(I + A)1/dL, s

)= ζ − sA ·

∑

l∈L\{o}

n ⊗ n

‖l‖s− s

2A2 ·

∑

l∈L\{o}

n ⊗ n

‖l‖s+

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖2s

− sζ

2d‖A‖2 + ζO(‖A‖3) as A → O, A ∈ T . (10)

Property (ii) has the following two consequences: First, the lattice L is stationary for arbitrarilylarge s. Thus, by Corollary 1 and Proposition 1,

each layer of L is strongly eutactic and, therefore, L is fully eutactic for each s > d. (11)

Second,

ζ

(I + A

det(I + A)1/dL, s

)≥ ζ as A → O, A ∈ T , for arbitrarily large s. (12)

Propositions (10)–(12) and Definition 2 together yield the following statement:

ζ − sζ

2d‖A‖2 +

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s− sζ

2d‖A‖2 ≥ ζ

as A → O, A ∈ T , for arbitrarily large s;

i.e.,∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s≥ 2ζ

d(s + 2)‖A‖2 for A ∈ T and arbitrarily large s.

Hence∑

l∈L1

(A · n ⊗ n)2 +

(‖l11‖s

‖l21‖s

∑

l∈L2

(A · n ⊗ n)2 + . . .

)≥ 4k1

d(s + 2)‖A‖2 +

4k2‖l11‖s

d(s + 2)‖l21‖s‖A‖2 + . . . .

Since the expression in large parentheses is less than the first term on the right-hand side if s issufficiently large, we have the following inequality:

∑

l∈L1

(A · n ⊗ n)2 > 0 for A ∈ T \ {O}.

By (11) the layer L1 is strongly eutactic. Hence, this inequality implies by Proposition 2(i) that L1

is perfect, concluding the proof of (iii).(iii) ⇒ (i). Since, by assumption, each layer is strongly eutactic, Proposition 1(i) shows that L

is fully eutactic for each s > d. Identity (5) then yields the following:

ζ

(I + A

det(I + A)1/dL, s

)= ζ − sζ

d‖A‖2 +

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s+ ζO(‖A‖3)

as A → O, A ∈ T , for each s > d. (13)



Since, again by assumption, L is perfect, we have∑

l∈L1

(A · n ⊗ n)2 > 0 for A ∈ T \ {O}.

This sum may be considered as a positive definite quadratic form in the variable A ∈ T . Hence,∑

l∈L1

(A · n ⊗ n)2 ≥ const · ‖A‖2 for A ∈ T .

Note that1

‖l11‖s≥ const · ζ for all sufficiently large s,

and therefore∑

l∈L1

(A · n ⊗ n)2

‖l‖s≥ ζ · const · ‖A‖2 for A ∈ T and all sufficiently large s. (14)

Relations (13) and (14) together show that

ζ

(I + A

det(I + A)1/dL, s

)≥ ζ +

s2

2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s− sζ

d‖A‖2 + ζO(‖A‖3)

≥ ζ + ζ · const · s2‖A‖2 as A → O, A ∈ T , for all sufficiently large s,

concluding the proof of (i). �Proof of Corollary 4. (i) By assumption, L is perfect and, by Proposition 2(iv), each layer

of L is strongly eutactic. Now apply Theorem 3 to see that L is quadratic minimum for all sufficientlylarge s.

(ii) Proposition 1(ii) implies that each layer of L is strongly eutactic. Since, by assumption, Lis also perfect, Theorem 3 shows that it is quadratic minimum for all sufficiently large s.

(iii) By Propositions 1(i) and 1(iii), the lattice L is both fully eutactic and completely eutacticfor each s > d. The definition of complete eutaxy and Theorem 2 then show that L is quadraticminimum for each s > d.

(iv) Apply Proposition 1(iii) and the proof of Corollary 4(iii). �Proof of Corollary 5. (i) Among the lattices described in Corollary 2(i) only those in Bravais

types hp and cF are perfect and all their layers are strongly eutactic. Now apply Theorem 3.(ii) This follows from Corollary 2(ii) and Theorem 3. �

3. MINIMUM PROPERTIES OF ζC

It turns out that many of the results of the previous section can be extended to lattice zetafunctions, where the Euclidean norm is replaced by a smooth norm.

Let C be a convex body, that is, a compact convex subset of Ed with non-empty interior. We

assume that C is o-symmetric and that its boundary is of class C1. Let ‖ · ‖C be the norm on Ed with

unit ball C and L a lattice with d(L) = 1. Define the lattice zeta function ζC(L, ·) correspondingto C and L by

ζC(L, s) =∑

l∈L\{o}

1‖l‖s

C

for s > d.

Using suitable variants of the Voronoı type notions of eutaxy and perfection, we characterizestationary lattices for given s > d (Theorem 4) and lattices which are quadratic minimum for allsufficiently large s (Theorem 5). A connection to lattice packings of C is mentioned (Corollary 7).


116 P.M. GRUBER

Minimum and Voronoı type notions. Let s > d. Stationarity, minimality, and quadraticminimality of ζC at a lattice L for s are defined as for ζ but with

S ={A ∈ E

d2: A · I = tr A = 0

}

instead of T , where Ed2 is identified with the space of all d × d matrices.

To introduce the needed versions of eutaxy, we assign to each l ∈ Ed \ {o} a pair of vectors, the

exterior normal (Euclidean) unit vector u of the smooth body ‖l‖CC at its boundary point l, andthe vector n = l/(l · u). Let u ⊗ n = unT. The points of L \ {o} with the same norm form a layerof L with respect to ‖ · ‖C . A layer M of L with 2k = #M is strongly eutactic with respect to C if

∑

l∈M

u ⊗ n =2kd

I.

If the first layer of L is strongly eutactic with respect to C, then the lattice L is said to be stronglyeutactic with respect to C. It is fully eutactic with respect to C and s if

∑

l∈L\{o}

u ⊗ n

‖l‖sC

=ζC

dI,

where ζC = ζC(L, s). We call M perfect with respect to C if

Ed2

= lin{u ⊗ n : l ∈ M}.

If the first layer of L is perfect with respect to C, then the lattice L is said to be perfect withrespect to C.

Characterization of stationarity and quadratic minimality. The proofs of the followingresults for ζC are similar to the proofs of the corresponding results for ζ and, thus, are omitted. ToTheorem 1 there corresponds the following criterion.

Theorem 4. Let s > d. Then the following properties of ζC and L are equivalent :(i) ζC is stationary at L for s;(ii) L is fully eutactic with respect to C and s.

Given C and s, there is a lattice with determinant 1 for which ζC(·, s) is minimum and thusstationary.

It is possible to extend Theorem 2. Since the extended inequality condition seems to be difficultto check, we do not state it. A partial extension of Theorem 3 is as follows.

Theorem 5. Let C be of class C2. Then the following properties of ζC and L are equivalent :(i) ζC is quadratic minimum at L for all sufficiently large s;(ii) L is perfect with respect to C and each layer of L is strongly eutactic with respect to C.

Relations to lattice packing of a convex body. The notions of lattice packing of C,density, the refined maximum properties of semi-stationarity and ultra-maximality of L with respectto the density and the Voronoı type properties of eutaxy and perfection of L with respect to C aredefined as in Section 2, with Bd and T replaced by C and S.

Results of the author [25] and Theorems 4 and 5 together yield the following result.Corollary 7. Among the minimum properties of the lattice zeta function ζC and the maximum

properties of the lattice packing density δ(C, ·) the following relations hold :(i) If ζC is stationary at L for arbitrarily large s, then δ(C, ·) is semi-stationary at L.(ii) If ζC is quadratic minimum at L for arbitrarily large s, then δ(C·) is ultra-maximum at L.



Tools for the proofs. We do not give proofs of these results, but state one tool.Proposition 3. The convex body C and the lattice L have the following properties :(i) If each layer of L is strongly eutactic with respect to C, then L is fully eutactic with respect

to C and each s > d.(ii) If L is fully eutactic with respect to C for arbitrarily large s, then each layer of L is strongly

eutactic with respect to C.(iii) If a layer M of L is strongly eutactic with respect to C and satisfies the inequality

∑

l∈M

(A · u ⊗ n)2 > 0 for A ∈ S \ {O},

then it is perfect with respect to C.

4. MINIMUM PROPERTIES OF ζζ∗

Let C be an o-symmetric, smooth, and strictly convex body and L a lattice in Ed, with d(L) = 1.

The polar body C∗ of C and the polar lattice L∗ of L are defined by

C∗ ={y ∈ E

d : x · y ≤ 1 for x ∈ C}, L∗ =

{m ∈ E

d : l · m ∈ Z for l ∈ L},

where Z is the ring of rational integers. The so defined C∗ is an o-symmetric, smooth, and strictlyconvex body and L∗ is a lattice with d(L∗) = 1.

There is a voluminous literature in which C, L and C∗, L∗ are considered simultaneously.See, for example, the investigation of the product of the packing densities δ(Bd, L)δ(Bd, L∗) andδ(C,L)δ(C∗ , L∗) by Berge and Martinet [7] and the author [25]. We also refer to the pertinentliterature in the geometry of numbers and the asymptotic theory of normed spaces.

In this section the product ζ(L, s)ζ(L∗, s) will be studied. We begin with the introduction ofthe necessary terminology and then investigate dual stationarity (Theorem 6 and Corollary 8) anddual quadratic minimality (Theorem 7 and Corollary 9).

Minimum and Voronoı type properties. In the following ζ and ζ∗ stand for ζ(L, s) andζ(L∗, s). Let s > d. The function ζ is dual stationary, dual minimum, or dual quadratic minimumat the lattice L for s if

ζ(

I+Adet(I+A)1/d L, s

)ζ((

I+Adet(I+A)1/d L

)∗, s

)

ζ(L, s)ζ(L∗, s)

⎧⎪⎨

⎪⎩

= 1 + o(‖A‖),≥ 1,

≥ 1 + const · s2‖A‖2

as A → O, A ∈ T . (15)

The needed Voronoı type notions are defined as follows. Let Li and Mi be the layers of L and L∗,respectively, and 2ki = #Li, 2k∗

i = #Mi, for i = 1, 2, . . . .Definition 5. A layer Li of L is dual strongly eutactic if

12ki

∑

l∈Li

l ⊗ l

‖l‖2=

12k∗

i

∑

m∈Mi

m ⊗ m

‖m‖2.

Definition 6. L is dual fully eutactic for s if it satisfies one of the two equivalent conditions

1ζ

∑

l∈L\{o}

l ⊗ l

‖l‖s+2=

1ζ∗

∑

m∈L∗\{o}

m ⊗ m

‖m‖s+2,

1ζ

∑

l∈L\{o}

A · l ⊗ l

‖l‖s+2=

1ζ∗

∑

m∈L∗\{o}

A · m ⊗ m

‖m‖s+2for A ∈ T .


118 P.M. GRUBER

Definition 7. A layer Li of L is dual ultra eutactic if

12ki

∑

l∈Li

(A · l ⊗ l)2

‖l‖4=

12k∗

i

∑

m∈L∗i

(A · m ⊗ m)2

‖m‖4for A ∈ T .

Definition 8. L is dual completely eutactic for s if

1ζ

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4=

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)2


Characterization of dual stationary lattices. In analogy to Theorem 1 we have the fol-lowing result.


(i) ζ is dual stationary at L for s;

(ii) L is dual fully eutactic for s.

Dual full eutaxy is difficult to check. The following result contains conditions which, sometimes,are more convenient to apply.

Corollary 8. The following conditions (i) and (ii) are necessary and sufficient for L in orderthat ζ be dual stationary at L for each s > d; condition (iii) is sufficient :

(i) L is dual fully eutactic for each s > d;

(ii) each layer of L is dual strongly eutactic;(iii) the first layer of L is perfect and A acts transitively on it.

Characterization of dual quadratic minimality. The next result is the dual version ofTheorem 2.


(i) ζ is dual quadratic minimum at L for s;

(ii) L is dual fully eutactic for s and satisfies the inequality

(s + 2)

(1ζ

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4+

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)2

‖m‖s+4

)

>4ζ

∑

l∈L\{o}

A2 · l ⊗ l

‖l‖s+2+

2sζ

(∑

m∈L∗\{o}

A · l ⊗ l

‖l‖s+2

)2

for A ∈ T \ {O}.

The following corollary may be easier to apply.Corollary 9. Each of the following conditions on L is sufficient for ζ to be dual quadratic

minimum at L for each s > d:

(i) L is completely eutactic and dual completely eutactic for each s > d;

(ii) each layer of L is ultra eutactic and dual ultra eutactic.

Tools for the proofs. The identity((I + A)L

)∗ =((I + A)T

)−1L∗ = (I − A + A2 − . . .)L∗ as A → O, A ∈ T ,



implies

(l + Al)2 = l2(1 + 2A · n ⊗ n + A2 · n ⊗ n

),

(m − Am + A2m − . . .

)2 = m2(1 − 2A · p ⊗ p + 3A2 · p ⊗ p + O(‖A‖3)

)

as A → O, A ∈ T , where l,m ∈ Ed \ {o}, n =

l

‖l‖ , p =m

‖m‖ ,

and thus

1‖l + Al‖s

=1

‖l‖s

(1 − sA · n ⊗ n − s

2A2 · n ⊗ n +

s(s + 2)2

(A · n ⊗ n)2 + O(‖A‖3))

,

1‖m − Am + . . . ‖s

=1

‖m‖s

(1 + sA · p ⊗ p − 3s

2A2 · p ⊗ p +

s(s + 2)2

(A · p ⊗ p)2 + O(‖A‖3))

as A → O, A ∈ T .

Hence,

ζ

(I + A

det(I + A)1/dL, s

)ζ

((I + A

det(I + A)1/dL

)∗, s

)= ζ

((I + A)L, s

)ζ(((I + A)L)∗, s

)

=

(ζ − sA ·

∑

l∈L\{o}

n ⊗ n

‖l‖s− s

2A2 ·

∑

l∈L\{o}

n ⊗ n

‖l‖s+

s(s + 2)2

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s+ ζO(‖A‖3)

)

×(ζ∗ + sA ·

∑

m∈L∗\{o}

p ⊗ p

‖m‖s− 3s

2A2 ·

∑

m∈L∗\{o}

p ⊗ p

‖m‖s+

s(s+2)2

∑

m∈L∗\{o}

(A · p ⊗ p)2

‖m‖s+ ζ∗O(‖A‖3)

)

= ζζ∗(1 − sA ·

(1ζ

∑

l∈L\{o}

n ⊗ n

‖l‖s− 1

ζ∗

∑

m∈L∗\{o}

p ⊗ p

‖m‖s

)− s

2A2 ·

(1ζ

∑

l∈L\{o}

n ⊗ n

‖l‖s+

3ζ∗

∑

m∈L∗\{o}

p ⊗ p

‖m‖s

)

+s(s + 2)

2

(1ζ

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s+

1ζ∗

∑

m∈L∗\{o}

(A · p ⊗ p)2

‖m‖s

)

− s2

ζζ∗

(∑

l∈L\{o}

A · n ⊗ n

‖l‖s

)(∑

m∈L∗\{o}

A · p ⊗ p

‖m‖2

)+ (ζ + ζ∗)O(‖A‖3)

)

as A → O, A ∈ T . (16)

The automorphism groups of L and L∗ are the same,

A(L) = A(L∗). (17)

We first show that A(L) ⊆ A(L∗). Let S ∈ A(L). Then SL = L and thus, by the definition of L∗,

SL∗ ={Sm : l · m ∈ Z for l ∈ L

}=

{Sm : Sl · Sm ∈ Z for l ∈ L

}

={Sm : l · Sm ∈ Z for l ∈ L

}=

{r : l · r ∈ Z for l ∈ L

}= L∗.

Hence S ∈ A(L∗), concluding the proof that A(L) ⊆ A(L∗). Applying this to L∗ instead of L andnoting that L∗∗ = L give A(L∗) ⊆ A(L∗∗) = A(L). Hence A(L) = A(L∗).


120 P.M. GRUBER

Proposition 4. The lattice L has the following properties :(i) If each layer of L is dual strongly eutactic or dual ultra eutactic, then L is dual fully eutactic

or dual completely eutactic, respectively, for each s > d.(ii) If L is dual fully eutactic or dual completely eutactic for arbitrarily large s, then each layer

of L is dual strongly eutactic or dual ultra eutactic, respectively.(iii) Let s > d. If L is dual completely eutactic for s, then it is also dual fully eutactic for s.

Proof. (i), (ii) The proofs of these statements are similar to those of the corresponding state-ments in Proposition 1.

(iii) By assumption,

1ζ

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4=

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)2


Differentiating each side of this identity with respect to A implies

1ζ

∑

l∈L\{o}

(A · l ⊗ l) l ⊗ l

‖l‖s+4=

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)m ⊗ m

‖m‖s+4.

Taking the traces of both sides of this identity shows that L is dual fully eutactic (see Defini-tion 6). �

Proofs of the theorems and corollaries.Proof of Theorem 6. (i) ⇔ (ii). In view of (16) and Definition 6, the proof of Theorem 6 is

simple:

ζ is dual stationary ⇔ ζ

(I + A

det(I + A)1/dL, s

)ζ

((I + A

det(I + A)1/dL

)∗, s

)= ζζ∗

(1 + o(‖A‖)

)

as A → O, A ∈ T

⇔ A ·(

1ζ

∑

l∈L\{o}

n ⊗ n

‖l‖s− 1

ζ∗

∑

m∈L∗\{o}

p ⊗ p

‖m‖s

)= 0 for A ∈ T

⇔ 1ζ

∑

l∈L\{o}

n ⊗ n

‖l‖s− 1

ζ∗

∑

m∈L∗\{o}

p ⊗ p

‖m‖s= αI for suitable α ∈ R

⇔ 1ζ

∑

l∈L\{o}

n ⊗ n

‖l‖s=

1ζ∗

∑

m∈L∗\{o}

p ⊗ p

‖m‖s

⇔ L is dual fully eutactic.

To see that α = 0, consider the traces of both sides. This gives 0 = 1 − 1 = αd. �Proof of Corollary 8. By Theorem 6 it is sufficient to show the implications (i) ⇔ (ii) ⇐ (iii).(i) ⇔ (ii). This follows from Proposition 4.(iii) ⇒ (ii). By Proposition 2(iv), each layer of L is strongly eutactic. Relation (17) and the

same proof as that of Proposition 2(iii) show that each layer of L∗ is strongly eutactic. Together,these statements imply that each layer of L is strongly dual eutactic (see Definition 5). �

Proof of Theorem 7. To see the equivalence of (i) and (ii), we proceed as follows:

ζ is dual quadratic minimum ⇔ A ·(

1ζ

∑

l∈L\{o}

n ⊗ n

‖l‖s− 1

ζ∗

∑

m∈L∗\{o}

p ⊗ p

‖m‖s

)= 0 for A ∈ T and



s(s + 2)2

(1ζ

∑

l∈L\{o}

(A · n ⊗ n)2

‖l‖s+

1ζ∗

∑

ml∈L∗\{o}

(A · p ⊗ p)2

‖m‖s

)

−s

2

(1ζA2 ·

∑

l∈L\{o}

n ⊗ n

‖l‖s+

3ζ∗

A2 ·∑

l∈L∗\{o}

p ⊗ p

‖m‖s

)

−s2

(1ζA ·

∑

l∈L\{o}

n ⊗ n

‖l‖s

)(1ζ∗

A ·∑

m∈L∗\{o}

p ⊗ p

‖m‖s

)+ O(‖A‖3)

≥ const · ‖A‖2 for A ∈ T (by (16) and Definition 6)

⇔ 1ζ

∑

l∈L\{o}

l ⊗ l

‖l‖s+2=

1ζ∗

∑

m∈L∗\{o}

m ⊗ m

‖m‖s+2(see the proof of Theorem 6) and

(s + 2)

(1ζ

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4+

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)2

‖m‖s+4

)

− 2sζ2

(∑

l∈L\{o}

A · l ⊗ l

‖l‖s+2

)2

− 4ζ

∑

l∈L\{o}

A2 · l ⊗ l

‖l‖s+2≥ const · ‖A‖2 as A → O, A ∈ T

⇔ L is dual fully eutactic for s and

(s + 2)

(1ζ

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4+

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)2

‖m‖s+4

)

>2sζ2

(∑

l∈L\{o}

A · l ⊗ l

‖l‖s+2

)2

+4ζ

∑

l∈L\{o}

A2 · l ⊗ l

‖l‖s+2for A ∈ T \ {O}. �

Proof of Corollary 9. (ii) ⇔ (i). This follows from Proposition 4.We have to show that (i) implies that ζ is dual quadratic minimum for each s > d. To see this,

note that by assumption each layer of L is completely eutactic and dual completely eutactic for eachs > d. By Propositions 1(iii) and 4(iii), L is fully eutactic and dual fully eutactic for any s > d.Thus,

1ζ

∑

l∈L\{o}

A · l ⊗ l

‖l‖s+2=

1ζ∗

∑

m∈L∗\{o}

A · m ⊗ m

‖m‖s+2= 0,

1ζ

∑

l∈L\{o}

A2 · l ⊗ l

‖l‖s+2=

1ζ∗

∑

m∈L∗\{o}

A2 · m ⊗ m

‖m‖s+2=

‖A‖2

d,

1ζ

∑

l∈L\{o}

(A · l ⊗ l)2

‖l‖s+4=

1ζ∗

∑

m∈L∗\{o}

(A · m ⊗ m)2

‖m‖s+4=

2‖A‖2

d(d + 2)for A ∈ T .

These equalities show that L is dual fully eutactic and satisfies the inequality in Theorem 7(ii) forany s > d. Now apply Theorem 7 to see that ζ is dual quadratic minimum at L for any s > d. �


122 P.M. GRUBER

5. MINIMUM PROPERTIES OF ζCζC∗

Let C be an o-symmetric, smooth, and strictly convex body and L a lattice with d(L) = 1in E

d. Let C∗ and L∗ be their polars. Our aim is to indicate minimum properties of the productζC(L, s) ζC∗(L∗, s). We state only a characterization of dual stationary lattices for a given s > d(Theorem 8). In this case it is clear that there are lattices which are dual stationary. It is possibleto prove characterizations of stationary lattices for all s > d or all sufficiently large s.

Minimum and Voronoı type properties. Let s > d. Dual stationarity of ζC at a lattice Lfor s is defined as for ζ in Section 4, but with S instead of T .

The Voronoı type property which characterizes lattices at which ζC is dual stationary for s isdual full eutaxy with respect to C and s:

1ζC

∑

l∈L\{o}

u ⊗ n

‖l‖sC

=1

ζC∗

∑

m∈L∗\{o}

p ⊗ v

‖m‖sC∗

for A ∈ T ,

where n and u are assigned to l as in Section 3 and, similarly, p and v to m. Note that on theright-hand side the factors of the tensor product are in reverse order.

Characterization of dual stationarity. Theorem 6 can be extended as follows.Theorem 8. Let s > d. The following properties of ζC and L are equivalent :(i) ζC is dual stationary at L for s;(ii) L is dual fully eutactic for s.

6. OPEN PROBLEMS

In connection with the results in this article, a series of problems arise. Three of them will bestated in the following. Let L be a lattice in E

d with d(L) = 1.The earlier characterizations and sufficient conditions for L guarantee in many cases that ζ or ζC

is stationary, minimum, or quadratic minimum at L for a given s > d, for all sufficiently large s, orfor all s > d. The problem is to make this family of results complete. One question is the following.

Problem 1. Characterize the lattices (if any) at which ζ is minimum (but not quadraticminimum) for a given s > d, for all sufficiently large s, or for all s > d.

In the following problems, “most” is to be understood in the sense of Baire categories (see, forexample, [19]).

Problem 2. Show that (in sufficiently high dimensions) for most o-symmetric convex bodies Cthere are lattices at which ζC(·, s) and ζC(·, s) ζC∗(·∗, s) are quadratic minimum for a given s > d,for all sufficiently large s, or for all s > d.

A positive answer to this problem would also settle the question of the existence of convex bodieswith eutactic and perfect lattices (cf. [25]).

Problem 3. Is it true that (in all sufficiently high dimensions) for most o-symmetric convexbodies C, the lattice L with d(L) = 1 for which ζC(·, s) attains its global minimum for a givens > d, for all sufficiently large s, or for all s > d has the following properties:

(i) L is unique;(ii) the global minimum is quadratic?

ACKNOWLEDGMENTS

For their help in the preparation of this article and for references to the literature, the authoris obliged to Iskander Aliev, Renaud Coulangeon, Jacques Martinet, and Tony Thompson.



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application of an idea of voronoĭ to lattice zeta functions

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