UPPSALA DISSERTATIONS IN MATHEMATICS

68

Asymptotic Problems on Homogeneous Spaces

Anders Södergren

Department of MathematicsUppsala UniversityUPPSALA 2010

Till mamma och pappa

List of Papers

This thesis is based on the following papers, which are referred to in thetext by their Roman numerals.

I Södergren, A. (2010) On the uniform equidistribution of closedhorospheres in hyperbolic manifolds. Manuscript.

II Södergren, A. (2010) On the Poisson distribution of lengths oflattice vectors in a random lattice. To appear in MathematischeZeitschrift.

III Södergren, A. (2010) On the distribution of angles between the Nshortest vectors in a random lattice. Manuscript.

IV Södergren, A. (2010) On the value distribution and moments of theEpstein zeta function to the right of the critical strip. Manuscript.

V Södergren, A. (2010) On the value distribution of the Epstein zetafunction in the critical strip. Manuscript.

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Hyperbolic n-space . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 The space of n-dimensional lattices . . . . . . . . . . . . . . . 2

1.2 Zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Epstein’s zeta function . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Summary of the papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1. Introduction

1.1 Homogeneous spacesIn this thesis we study two special families of homogeneous spaces,namely hyperbolic n-space and the space of n-dimensional lattices. Thesespaces are special in the sense that they have more structure than isneeded to be a homogeneous space. Nevertheless, let us start by givingthe definition of a (general) homogeneous space.Formally, a homogeneous space is a pair (X, ρ) of a set X and a tran-

sitive action ρ on X by a group G. Let us assume for the moment thatρ is a left action. When G and ρ are clear from the context, we willadopt the common practice and denote the homogeneous space simplyby X and write the action of g ∈ G on X as x 7→ g(x). If the set X hasmore structure it is natural to demand that the action ρ preserves thisstructure. In particular, if X is a smooth manifold we demand that thegroup G acts on X by diffeomorphisms.Every homogeneous space can in a natural but non-canonical way

be realized as a space of cosets as follows. Fix any x0 ∈ X and letGx0 =

g ∈ G | g(x0) = x0

be the stabilizer of x0. Then the set of left

cosets G/Gx0 can be identified with X via the map ηx0 : G/Gx0 → Xgiven by gGx0 7→ g(x0).We will be interested in the following special case. LetX be a Hausdorff

topological space and G a Lie group acting transitively and continuouslyon X. Here the last condition means that (g, x) 7→ g(x) is a continuousmapping of G × X onto X. Now, given x0 ∈ X we find that Gx0 is aclosed subgroup of G and hence [9, Thm. 4.2, p. 123] implies that X hasa unique smooth structure such that (g, x) 7→ g(x) is a smooth mappingand ηx0 is a diffeomorphism.Many common spaces can be realized as homogeneous spaces of the

above type. To mention a few examples; the unit sphere Sn−1 ⊂ Rn isdiffeomorphic to the homogeneous space SO(n)/SO(n− 1), the standardn-dimensional torus Tn is diffeomorphic to Rn/Zn and the real projectivespace RPn−1 is diffeomorphic to SO(n)/O(n − 1) (where O(n− 1) isconsidered as a subgroup of SO(n) in an appropriate way, see [30, Sec.3.65]).

1

1.1.1 Hyperbolic n-spaceThe n-dimensional hyperbolic space is the unique n-dimensional simplyconnected Riemannian manifold with constant sectional curvature equalto −1. It is one of the standard examples of an n-dimensional space withnon-Euclidean geometry. A commonly used model of hyperbolic n-spaceis the set

Hn =x = (x1, x2, . . . , xn+1) ∈ Rn+1

∣∣x21 − x2

2 − . . .− x2n+1 = 1, x1 > 0

equipped with the Riemannian metric induced from the pseudo-Riemannian metric

ds2 = −dx21 + dx2

2 + . . .+ dx2n+1

on Rn+1. The full group of orientation preserving isometries of Hn isgiven by the Lie group PSO(1, n) (see e.g. [13, Sec. 3.2 (Cor. 3)]). It isnow easy to verify that the stabilizer of (1, 0, . . . , 0) ∈ Hn equals

1 0 . . . 0

0... A

0

∈ PSO(1, n) : A ∈ SO(n)

,

which is clearly isomorphic to SO(n). Hence we find that Hn is diffeo-morphic to the homogeneous space PSO(1, n)/SO(n).We remark that there are several other important models of hyperbolic

n-space. In Paper I we consider the upper half-space model Hn since it isin many ways convenient in connection with the problem we study. Fora description of Hn see Section 2.1 and Paper I.

1.1.2 The space of n-dimensional latticesBy definition, an n-dimensional lattice L is the Z-span of a basisv1, . . . ,vn in Rn, i.e.

L = n∑i=1

aivi∣∣ ai ∈ Z, 1 ≤ i ≤ n

.

The set of points

P = n∑i=1

tivi∣∣ 0 ≤ ti < 1, 1 ≤ i ≤ n

is called a fundamental parallelogram (or a fundamental region) for thelattice L. Note that the set P is naturally identified with the torus Rn/L.The covolume of L is defined as the n-dimensional Euclidean volume of

2

P (or equivalently, the volume of Rn/L). We let Xn denote the space ofn-dimensional lattices of covolume 1.Note that SL(n,R) acts transitively from the right on Xn by the map-

ping (L, g) 7→ Lg, where of course

Lg =xg | x ∈ L

.

The fact that SL(n,Z) is the stabilizer of Zn ∈ Xn implies that the el-ements of Xn can be identified with the elements of the homogeneousspace SL(n,Z)\SL(n,R). Note in particular that this identification in-duces a smooth structure on Xn.As any locally compact (Hausdorff) topological group, SL(n,R) is

equipped with a right Haar measure (cf. [2, Chap. 9]). In fact, sinceSL(n,R) is even a semisimple Lie group, any right Haar measure is alsoa left Haar measure and we can thus without ambiguity speak about aHaar measure on SL(n,R) (cf. [11, Sec. VIII.2]). Recall that a right (left)Haar measure is a non-zero regular Borel measure that is invariant underright (left) translations. When such a measure exists it is unique up tomultiplication with a positive constant. We can define a Haar measureon SL(n,R) as the measure νn on SL(n,R) which satisfies

dνn(M ′)dt

t=(

det(xij))−n

dx11dx12 . . . dxnn, (1.1.1)

when parametrizing GL+(n,R) as (xij)ni,j=1 = t1/nM ′, with

M ′ ∈ SL(n,R) and t > 0. Let us note that, in more concrete terms, themeasure νn is given by the following explicit formula, for any givenBorel subset A ⊂ SL(n,R):

νn(A) = limε→0

ε−1ν ′n

(M ∈ GL+(n,R)

∣∣ 1 ≤ detM ≤ 1 + ε, π(M) ∈ A),

where ν ′n is the measure on GL+(n,R) given by (1.1.1) andπ : GL+(n,R) → SL(n,R) is the natural projection given byπ(M) = (detM)−1/nM .By restricting νn to a fundamental region Fn ⊂ SL(n,R) for

SL(n,Z)\SL(n,R) we can induce a measure µn also on Xn. Note inparticular that this measure is invariant under right multiplication withelements in SL(n,R). It is of great importance in this thesis that µn, orequivalently the restriction of νn to Fn, is a finite measure. For a proofof this result we refer to Raghunathan [12, Cor. 10.5]. However, forn ≥ 2 we note that it is possible to give a more precise result. In fact, ifwe define ν ′′n to be the (non-Haar) measure on GL+(n,R) given by

dν ′′n(M) = dx11dx12 . . . dxnn,

3

it is straightforward to prove that for any measurable set A ⊂ SL(n,R)we have the relation

νn(A) = n · ν ′′n(M ∈ GL+(n,R)

∣∣ 0 < detM ≤ 1, π(M) ∈ A).

(1.1.2)

Furthermore, if we let

n =M ∈ GL+(n,R)

∣∣ 0 < detM ≤ 1, π(M) ∈ Fn

it follows from [20, Sec. XV] that

ν ′′n( n) =ζ(2)ζ(3) · · · ζ(n)

n,

where ζ(s) is the Riemann zeta function. Thus it follows from (1.1.2)that

µn(Xn) = νn(Fn) = ζ(2)ζ(3) · · · ζ(n).

Finally we normalize µn as

µn =(ζ(2)ζ(3) · · · ζ(n)

)−1 · µn,

in order to turn it into a probability measure on Xn. The probabilityspace (Xn, µn) is the scene for the investigations presented in PapersII,III,IV and V.

1.2 Zeta functionsOne of the most famous and important functions in all of mathematicsis the Riemann zeta function. It is defined as the absolutely convergentsum

ζ(s) =

∞∑n=1

1

ns(1.2.1)

for Re s > 1. It is well-known that ζ(s) has an analytic continuation toC\1 and that s = 1 is a simple pole with residue 1. In this connectionwe note that ζ(s) satisfies a functional equation. More precisely we have

ξ(s) = ξ(1− s), (1.2.2)

where

ξ(s) = π−s/2Γ(s/2)ζ(s).

4

The relation between ζ(s) and number theory origins from the so calledEuler product formula

ζ(s) =∏p

(1− 1

ps

)−1

, (1.2.3)

which holds for Re s > 1. The product on the right hand side in (1.2.3) istaken over the set of all prime numbers. This remarkable formula relatesthe additive structure of the positive integers on the left hand side to themultiplicative structure of the integers and primes on the right hand side.It turns out that even though (1.2.3) only holds for Re s > 1, the behaviorof ζ(s) in the critical strip 0 < Re s < 1 has important implications inthe theory of prime numbers. In particular there is a close connection(as we will se parts of below) between the complex numbers s satisfyingζ(s) = 0 and the prime counting function π(x) = #p prime | p ≤ x.Studying the symmetry coming from the functional equation (1.2.2)

it is easy to see that ζ(s) has zeros at the negative even integers−2,−4,−6, . . .. These are the so called trivial zeros. The Riemannhypothesis asserts that all the remaining zeros of ζ(s) are located onthe (critical) line Re s = 1

2 . If true the hypothesis would among otherthings imply that the counting function π(x) is very well approximatedby the logarithmic integral

Li(x) =

∫ x

2

dt

log t

in the sense that

π(x) = Li(x) +O(x1/2+ε

)(1.2.4)

would hold as x→∞ (cf. [10]). The estimate in (1.2.4) is in fact equiva-lent to the Riemann hypothesis. Despite numerous attempts and massivenumerical evidence in its favor, the Riemann hypothesis has not yet beenproved or disproved. It is widely considered as one of the most importantunsolved problem in modern mathematics.Functions defined in various contexts (possibly very different from

prime number theory) that are in one way or another analogous to ζ(s)are called zeta functions or in some cases L-functions. One example is theDedekind zeta function of a number field k, arising in algebraic numbertheory. Let k denote the ring of integers in k. For each non-zero idealof k, the norm of , N( ), is defined as the cardinality of the quotientring k/ . Now the Dedekind zeta function of k is defined for Re s > 1by the series

ζk(s) =∑⊆ k

1

N( )s,

5

where the sum is over all non-zero ideals of k. Note that this formulareduces to (1.2.1) when k = Q. For each number field k the associatedzeta function ζk has an analytic continuation to C \ 1 and it satisfiesa functional equation similar to (1.2.2) relating ζk(s) to ζk(1 − s). Fur-thermore, using the unique factorization of ideals (into prime ideals) ink and the multiplicativity of the norm N , one can for Re s > 1 write

also ζk as an Euler product

ζk(s) =∏⊆ k

(1− 1

N( )s

)−1

,

where the product is over all prime ideals of k. It follows from thisformula that ζk(s) also for general number fields k is closely related tothe study of prime ideals in k. Finally we mention that there exists anextended Riemann hypothesis asserting that for every number field k allzeros of ζk(s) with 0 < Re s < 1 are in fact on the line Re s = 1

2 .

1.2.1 Epstein’s zeta functionLet n ∈ Z+ be given. For lattices L ∈ Xn and s ∈ C with Re s > n

2 wedefine the Epstein zeta function by

En(L, s) =∑m∈L

′|m|−2s, (1.2.5)

where ′ denotes that the zero vector should be omitted. These functions(actually a slightly more general family of functions) were introducedby Epstein in the papers [3] and [4] in an attempt to find the mostgeneral form of a function satisfying a functional equation of the sametype (1.2.2) as the Riemann zeta function. Exploring the relationshipbetween En(L, s) and the lattice theta function, defined for Im z > 0 by

Θ(L, z) =∑m∈L

eπiz|m|2,

Epstein proved that En(L, s) has an analytic continuation to C except fora simple pole at s = n

2 with residue πn/2Γ(n2 )−1. Furthermore En(L, s)satisfies the functional equation

Fn(L, s) = Fn(L∗, n2 − s), (1.2.6)

where

Fn(L, s) = π−sΓ(s)En(L, s),

and L∗ is the dual lattice of L.

6

As can be seen above, En(L, s) is in many ways analogous to theRiemann zeta function. In particular we have the relation

ζ(2s) =1

2E1(Z, s).

The Epstein zeta function is further related to the Dedekind zeta func-tion via Hecke’s integral formula (cf. [7, pp. 198-207]). Note however thatEn(L, s) in general cannot be written as an Euler product. We also men-tion that for all n ≥ 2 there exist lattices L ∈ Xn for which the Riemannhypothesis for En(L, s) is known to fail in the sense that En(L, s) haszeros in the critical strip 0 < Re s < n

2 which are not on the line Re s = n4

(cf. [27, Thm. 1]; see also [1], [22], [26] and [28]).Finally we remark that the Epstein zeta function can alternatively be

defined in terms of positive definite quadratic forms in n variables. In thissetting it is also possible to consider generalizations of the Epstein zetafunction twisted with some character (cf. [23],[24]). It is a curious factthat such character sums are, in the case with binary quadratic forms,related to Gauss’ class number problem for imaginary quadratic fields.In particular Stark considers a particular instance of such a charactersum in his proof [21] that there exist exactly nine imaginary quadraticfields of class-number one. For a historical account on the class numberproblem as well as some indications of its relation to the Epstein zetafunction see [6].

7

2. Summary of the papers

In this chapter we give a short description of the content of each paperincluded in the thesis.

2.1 Paper IIn Paper I we study the asymptotic equidistribution of pieces of largeclosed horospheres in a cofinite hyperbolic manifold M of arbitrary di-mension. To present the problem, let us first consider the case when Mis two dimensional. Then the surface M has a finite number of cusps,and to each cusp corresponds a one-parameter family of closed horocy-cle curves in M . In each family there exists a unique closed horocycleof any given length ` > 0, and it is known that as ` → ∞, the closedhorocycle becomes asymptotically equidistributed on M with respect tothe hyperbolic area measure. Investigations related to this fact have beencarried out by a number of people over the years, including Selberg (un-published), Zagier [31], Sarnak [16], Hejhal [8], Flaminio and Forni [5],and Strömbergsson [25].In [8], Hejhal asked to what exact degree of uniformity does this

equidistribution hold? Specifically, given a subsegment of length `1 < `of a closed horocycle of length `, under what conditions on `1 can weensure that this subsegment becomes asymptotically equidistributed onM? Hejhal proved that this holds as long as we keep `1 ≥ `c+ε (ε > 0)as `→∞, where c ≥ 2

3 is a constant which only depends on the surfaceM . This was later improved in [25] to allowing c = 1

2 , independently ofM . This constant is optimal. The equidistribution results both in [8] and[25] were obtained with explicit rates.The purpose of Paper I is to generalize these equidistribution results

to the case when M is a hyperbolic manifold of arbitrary dimensionn + 1. We realize M as the quotient M = Γ \ Hn+1, where Hn+1 is thehyperbolic upper half space,

Hn+1 =P = (x, y)

∣∣ x ∈ Rn, y ∈ R>0

with Riemannian metric ds2 = y−2(dx2

1 + . . . + dx2n + dy2), and Γ is a

cofinite (but not cocompact) discrete subgroup of the group G of orien-tation preserving isometries of Hn+1. Without loss of generality we canassume that one of the cusps is placed at infinity. Then the fixator group

9

Γ∞ ⊂ Γ contains a subgroup of finite index consisting of translations,

Γ′∞ =

(x, y) 7→ (x+ ω, y)∣∣ ω ∈ Λ

where Λ is some lattice in Rn. Let Ω = ω1,ω2, . . . ,ωn be a basis of Λ.Now, for each y > 0 and any αi, βi ∈ R with αi < βi, we set

=

(u1ω1 + · · ·+ unωn, y)∣∣ ui ∈ [αi, βi] for i = 1, . . . , n

.

This is a box-shaped subset of a closed horosphere in M . Our firstmain theorem says that as y → 0, the box becomes asymptoticallyequidistributed in M with respect to the hyperbolic volume measuredν = y−n−1dx1 . . . dxndy, so long as we keep all βi − αi ≥ y1/2−ε.

Theorem 1. Let Γ be a cofinite discrete subgoup of G such that Γ\Hn+1

has a cusp at infinity. Let ε > 0, and let f be a fixed continuous functionof compact support on Γ \Hn+1. Then

1

(β1 − α1) · · · (βn − αn)

∫ β1

α1

· · ·∫ βn

αn

f(u1ω1 + · · ·+ unωn, y) du1 . . . dun

→ 1

ν(Γ \Hn+1)

∫Γ\Hn+1

f(P ) dν(P ),

(2.1.1)

uniformly as y → 0 so long as β1 − α1, . . . , βn − αn ≥ y1/2−ε.

Note that when n = 1 this specializes to the equidistribution resultfrom [25] described above, since ` ∼ y−1 in this case. The exponent 1

2in Theorem 1 is in fact the best possible in arbitrary dimension, if werestrict to considering boxes which have all side lengths comparable, asy → 0.We obtain all equidistribution results in Paper I with precise rates,

provided that f is sufficiently smooth. In this direction we obtain thebest results if instead of the box we use a smooth cutoff function inthe closed horosphere. Thus let us fix χ : Rn → R to be a smooth functionof compact support. Given δ1, . . . , δn ∈ (0, 1] and γ = (γ1, . . . , γn) ∈ Rnwe define

χδ,γ(u) = χ(u1−γ1

δ1, . . . , un−γnδn

),

and consider the horosphere integral1

δ1 · · · δn

∫Rn

χδ,γ(u)f(u1ω1 + · · ·+ unωn, y) du,

where du = du1 . . . dun. Note that if we relax the condition that χ besmooth, and take χ to be the characteristic function of the unit cube

10

[−12 ,

12 ]n, then we get back the left hand side of (2.1.1) with αi = γi−δi/2

and βi = γi + δi/2.The precise rate of equidistribution which we obtain depends on the

spectrum of the Laplace-Beltrami operator ∆ on M . If there are smalleigenvalues 0 < λ < n2/4 of −∆ on M , then we take σ1 ∈ (n2 , n) so thatλ = σ1(n − σ1) is the smallest non-zero eigenvalue; otherwise, if thereare no small eigenvalues, we set σ1 = n

2 .

Theorem 2. Let Γ be a cofinite discrete subgroup of G such that Γ\Hn+1

has a cusp at infinity. Let ε > 0, let f be a fixed C∞-function on Hn+1

which is Γ-invariant and of compact support on Γ \ Hn+1, and let χ :Rn → R be a smooth function of compact support. Then, uniformly overall γ = (γ1, . . . , γn) ∈ Rn and all δ1, . . . , δn satisfying √y ≤ δ1, . . . , δn ≤1,

1

δ1 · · · δn

∫Rn

χδ,γ(u)f(u1ω1 + · · ·+ unωn, y) du

=〈χ〉

ν(Γ \Hn+1)

∫Γ\Hn+1

f(P ) dν(P ) +O(y−ε(y/δ2

min

)n−σ1), (2.1.2)

where σ1 ∈ [n2 , n) is as above, 〈χ〉 =∫Rn χ(x) dx, δmin = min

i∈1,...,nδi, and

the implied constant depends on Γ, f , χ, ε and Ω.

Our method of proof of these results is to use the spectral decompo-sition of the Laplace-Beltrami operator on M , involving the theory ofEisenstein series. To obtain our results we develop several bounds onsums over the Fourier coefficients of the individual eigenfunctions of M ,which are also of independent interest. In particular we prove a preciseRankin-Selberg type bound and bounds on coefficient sums twisted withan additive character.

2.2 Paper IIIn Paper II we study the distribution of lengths of non-zero lattice vectorsin a random lattice of large dimension. Given a lattice L ∈ Xn, weorder its non-zero vectors by increasing lengths as ±v1,±v2,±v3, . . .,set `j = |vj | (thus 0 < `1 ≤ `2 ≤ `3 ≤ . . .), and define

Vj =πn/2

Γ(n2 + 1)`nj ,

so that Vj is the volume of an n-dimensional ball of radius `j . We alsolet, for t ≥ 0,

Nt(L) = #j | Vj ≤ t

.

11

The first result concerning the statistics of the sequence `j∞j=1 is dueto Rogers. In [15] he shows that, for fixed t, Nt(L) has a distributionwhich converges weakly to the Poisson distribution with mean t

2 as n→∞. Clearly Rogers’ result determines the limit distribution of each fixed`j as n→∞. In particular the volume V1 has an exponential distributionwith expectation value 2 as n → ∞. In [19] Schmidt presents a moredirect study of the distribution of `1 using another method resulting inbetter error bounds as n→∞.The results of Rogers and Schmidt suggest very naturally that it should

be possible to understand not only the distribution of each fixed `j butalso the joint distribution of the entire sequence `j∞j=1 for a randomlattice L ∈ Xn as n → ∞. The purpose of Paper II is to point out thatthis is possible using Rogers’ methods. Our main theorem states that, asn→∞, the volumes Vj∞j=1 behave like the points of a Poisson processon the positive real line with intensity 1

2 .

Theorem 3. Let N(t), t ≥ 0 be a Poisson process on the positive realline with intensity 1

2 . Then the stochastic process Nt(·), t ≥ 0 convergesweakly to N(t), t ≥ 0 as n→∞.

There is a close conceptual relation between Theorem 3 and the mainresult (Theorem 1.1) by VanderKam in [29] (cf. also Sarnak [17]), whichstates that for a generic fixed lattice L ∈ Xn, the local spacing statisticsof the infinite sequence Vj∞j=1 exhibit Poissonian behavior (to a largerextent the larger the dimension). VanderKam’s theorem is of particularinterest in the context of the Berry-Tabor conjecture (cf. [29, Cor. 1.2]).In this direction, let us note that Theorem 3 can be viewed as a kindof “low-eigenvalue-high-dimension” analog of the Berry-Tabor conjecturefor flat tori. Indeed, Theorem 3 can be reformulated as follows.

Theorem 4. For a random flat torus Rn/L with L ∈ Xn the distribu-tion of the non-zero eigenvalues (normalized to have mean-spacing one)converges to the distribution of the points of a Poisson process on thepositive real line with intensity one as n→∞.

2.3 Paper IIIIn this paper we continue the study in Paper II of statistical propertiesof the non-zero lattice vectors ±v1,±v2,±v3, . . . in a random lattice oflarge dimension. The purpose of Paper III is to determine the distribu-tion of lengths and relative positions of the vectors ±v1, . . . ,±vN for arandom lattice L ∈ Xn, for N fixed and n→∞.We remark that, since the vectors vj∞j=1 are determined only up to

sign, the angles between them are a priori not well-defined. We avoidthis ambiguity by introducing a "symmetrized" angle measure ϕ, takingvalues in the interval [0, π2 ]. To be more specific, we denote the Euclidean

12

angle between the vectors x1,x2 ∈ Rn \ 0 by φ(x1,x2) and define

ϕ(x1,x2) =

φ(x1,x2) if φ(x1,x2) ∈ [0, π2 ],π − φ(x1,x2) otherwise.

Given L ∈ Xn and i, j ∈ Z≥1, we let ϕij = ϕ(vi,vj). Using Rogers’ meanvalue formula [14] we show that for a random lattice L ∈ Xn the anglesϕiji<j accumulate to π

2 , as n → ∞, with a rate at least comparablewith n−

12 . This fact suggests that it is natural to study the normalized

variables

ϕij =√n(π2 − ϕij

)=√n(π2 − ϕ(vi,vj)

).

The main result in Paper III is the following theorem, where we use theterm positive Gaussian variable to denote a random variable Φ satisfyingΦ = |X| for a random variable X ∈ N(0, 1).

Theorem 5. Let N ∈ Z≥2. The joint distribution of V1, . . . ,VN and ϕij,1 ≤ i < j ≤ N , converges, as n → ∞, to the joint distribution of thefirst N points of a Poisson process on the positive real line with intensity12 and a collection of

(N2

)independent positive Gaussian variables (which

are also independent of the first N variables).

Paper III concludes with a short discussion of the limit distribution ofany fixed number of successive minima of a random lattice L ∈ Xn asn→∞. We show that, for each N ∈ Z≥1, the first N successive minima(suitably normalized) has the same limit distribution as the N -tuple(V1, . . . ,VN ) as n→∞.

2.4 Paper IVIn Paper IV we study the asymptotic value distribution of the the Ep-stein zeta function En(L, s) for s > n

2 and a random lattice L ∈ Xn oflarge dimension n. This paper is mainly motivated by [18, Sec. 6], whereSarnak and Strömbergsson study the height function for flat tori in largedimensions. For the flat torus Rn/L, with L ∈ Xn, the height functionis given by

hn(Rn/L) = 2 log 2π +∂

∂sEn(L∗, s)|s=0.

Sarnak and Strömbergsson show that hn concentrates at the valuelog 4π − γ + 1 1 as n → ∞. Interpreted in terms of the Epstein zetafunction this result states that if ε > 0 is fixed then

ProbµnL ∈ Xn

∣∣∣ ∣∣ ∂∂sEn(L, s)|s=0 − (1− γ − log π)∣∣ < ε

→ 1 (2.4.1)

1Here γ is Euler’s constant.

13

as n → ∞. Note that (2.4.1) together with En(L, 0) = −1 (∀L ∈ Xn)give a fairly precise description of the behavior of En(L, s) at the points = 0 for most L ∈ Xn when n is large.The results in Paper IV give information about the value distribution

of En(L, s) for s > n2 with large n. The main result is that the value dis-

tribution, as n→∞, of the (suitably normalized) Epstein zeta functioncan be completely described in terms of the points of a Poisson processon the positive real line.

Theorem 6. Let Vn denote the volume of the n-dimensional unit ball.Let P be a Poisson process on the positive real line with intensity 1

2 andlet T1, T2, T3, . . . denote the points of P ordered so that 0 < T1 < T2 <T3 < · · · . Then, for fixed c > 1

2 , the distribution of the random variableV −2cn En(·, cn) converges to the distribution of 2

∑∞j=1 T

−2cj as n → ∞.

In fact, for any m ≥ 1 and fixed 12 < c1 < · · · < cm, the distribution of

the random vector(V −2c1n En(·, c1n), . . . , V −2cm

n En(·, cmn))

converges to the distribution of(2

∞∑j=1

T−2c1j , . . . , 2

∞∑j=1

T−2cmj

)as n→∞.

We can strengthen the result in Theorem 6 by showing that we do notonly have convergence in distribution but also convergence in momentsafter an explicit and tractable truncation. (A truncation is necessaryalready in order for the moments of En(·, s) to exist.)

Theorem 7. Let c > 12 and δ > 0 be fixed. Let I(·) be the indicator

function and let En,δ(L, s) be the truncation of En(L, s) that discardsthe contribution to En(L, s) from all lattice vectors in L belonging tothe n-ball of volume δ centered at the origin. Then every moment ofV −2cn En,δ(·, cn) converges to the corresponding moment of 2

∑∞j=1 I

(Tj >

δ)T−2cj as n→∞. Furthermore, for any m ≥ 1 and fixed 1

2 < c1 < · · · <cm, the joint moments of the random vector(

V −2c1n En,δ(·, c1n), . . . , V −2cm

n En,δ(·, cmn))

converge to the corresponding joint moments of(2

∞∑j=1

I(Tj > δ

)T−2c1j , . . . , 2

∞∑j=1

I(Tj > δ

)T−2cmj

)as n→∞.

14

Paper IV concludes with a discussion of the random function c 7→V −2cn En(·, cn) for c ∈ [A,B] with 1

2 < A < B. Using the fact that c 7→V −2cn En(L, cn) is convex for each fixed lattice L ∈ Xn, we prove a result

analogous to Theorem 6 determining the asymptotic value distributionof this C

([A,B]

)-valued random function.

2.5 Paper VIn Paper V we study the asymptotic value distribution of the Epsteinzeta function En(L, s) for 0 < s < n

2 and a random lattice L ∈ Xn

of large dimension n. More precisely, we give information on the valuedistribution of En(L, s) for n4 < s < n

2 with large n; using (1.2.6) it is theneasy to infer results also for the interval 0 < s < n

4 . In order to state ourfirst main theorem we introduce some notation. We consider a Poissonprocess P =

N(V ), V ≥ 0

on the positive real line with constant

intensity 12 , and let T1, T2, T3, . . . denote the points of the process ordered

in such a way that 0 < T1 < T2 < T3 < . . .. We let N(V ) = 2N(V ) anddefine, for all V ≥ 0,

R(V ) = N(V )− V. (2.5.1)

Finally we let Vn denote the volume of the unit ball in Rn.

Theorem 8. Let 14 < c1 < c2 <

12 . For each n ∈ Z≥1 consider

c 7→ V −2cn En(·, cn)

as a random function in C([c1, c2]

). Then the distribution of this random

function converges to the distribution of

c 7→∫ ∞

0

V −2c dR(V )

as n→∞.

A crucial ingredient in the proof of Theorem 8 is our result Theorem 3(in Paper II) on the distribution of lengths of lattice vectors in a randomlattice L ∈ Xn. In view of that result the following definitions are natural.Given L ∈ Xn and V ≥ 0 we let Nn(V ) denote the number of non-zerolattice points of L in the closed n-ball of volume V centered at the origin,and define

Rn(V ) = Nn(V )− V.

A second crucial ingredient in our proof of Theorem 8 is the followingbound on Rn(V ).

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Theorem 9. For all ε > 0 there exists Cε > 0 such that for all n ≥ 3and C ≥ 1 we have

ProbµnL ∈ Xn

∣∣ |Rn(V )| ≤ Cε(CV )12 (log V )

32

+ε, ∀V ≥ 10≥ 1− C−1.

Theorem 9 is interesting not only for being an important technicalpart of the proof of Theorem 8, but also for its connection with thefamous circle problem generalized to dimension n and general ellipsoids.Given V > 0, n ≥ 2 and L ∈ Xn the problem asks for the numberN (V ) = 1 +Nn(V ) of lattice points of L in the closed n-ball of volumeV centered at the origin. It is well-known that N (V ) is asymptotic tothe volume V of this ball. Hence 1+Rn(V ) equals the remainder term inthis asymptotic relation, and Theorem 9 implies that this remainder is V

12 (log V )

32

+ε as V →∞, for almost every L ∈ Xn. As far as we areaware, this fact has not been pointed out previously in the literature.In the last section of Paper V we extend the result in Theorem 8 to

the case c2 = 12 . In order for this to make sense we have to subtract the

singular part of V −2cn En(·, cn) from both the random functions appear-

ing in Theorem 8. As an application we prove the following theorem,strengthening a result [18, Thm. 3] by Sarnak and Strömbergsson on theasymptotic value distribution of the height function hn.

Theorem 10. The random variable

n(hn(L)−

(log(4π)− γ + 1

))+ log n

converges in distribution to

2 limc→ 1

2−

(∫ ∞0

V −2c dR(V ) +1

1− 2c

)− log π − 1

as n→∞.

Another application of Theorem 8 (and its generalization allowing c2 =12) concerns a question posed by Sarnak and Strömbergsson in [18]. Theynote that if there exists a lattice L0 ∈ Xn satisfying En(L, s) ≥ En(L0, s)for all 0 < s < n

2 and all L ∈ Xn then En(L0, s) < 0 for 0 < s < n2 .

Hence, for such a lattice L0, En(L0, s) has no zeros in (0,∞). Sarnak andStrömbergsson point out that it is an interesting question as to whetheror not lattices L ∈ Xn with En(L, s) 6= 0, ∀s > 0, do exist for all n (orall large n). In this direction we prove the following result.

Corollary 11. For any fixed 14 < c1 < c2 ≤ 1

2 , the limit

limn→∞

ProbµnL ∈ Xn

∣∣En(s, L) < 0 for all s ∈ [c1n, c2n] \ 12n

16

exists, and equals

f(c1, c2) := Prob∫ ∞

0

V −2c dR(V ) < 0 for all c ∈ [c1, c2] \ 12.

Moreover, for all 14 < c1 < c2 ≤ 1

2 the probability f(c1, c2) satisfies0 < f(c1, c2) < 1.

In particular, for any given ε > 0 the probability that

En(s, L) < 0 for all s ∈(0, (1

4 − ε)n]∪[(1

4 + ε)n, 12n)

holds tends to a positive limit as n→∞!

17

Summary in Swedish

Den här avhandlingen behandlar olika asymptotiska problem på tvåspeciella familjer av homogena rum, närmare bestämt de n-dimensionellahyperboliska rummen och rummen av n-dimensionella gitter. För desenare rummen inför vi beteckningen Xn.I Artikel I studerar vi den asymptotiska likafördelningen av delar av

slutna horosfärer i en icke-kompakt hyperbolisk mångfald M av ändligvolym och godtycklig dimension n + 1. Vi beskriver M som en kvot avtypen M = Γ \Hn+1, där

Hn+1 =

(x, y)∣∣ x ∈ Rn, y ∈ R>0

betecknar övre halvrumsmodellen för det (n + 1)-dimensionella hyper-boliska rummet och Γ är en diskret delgrupp till gruppen G av orien-teringsbevarande isometrier av Hn+1. Vi kan utan inskränkning anta attΓ har en spets i oändligheten. Då innehåller stabilisatorn Γ∞ ⊂ Γ endelgrupp av ändligt index bestående av translationer,

Γ′∞ =

(x, y) 7→ (x+ ω, y)∣∣ ω ∈ Λ

där Λ är något gitter i Rn. Låt Ω = ω1,ω2, . . . ,ωn vara en bas i Λoch sätt, för varje y > 0 och alla αi, βi ∈ R sådana att αi < βi,

=

(u1ω1 + · · ·+ unωn, y)∣∣ ui ∈ [αi, βi] för i = 1, . . . , n

.

Ett av huvudresultaten i Artikel I säger att då y → 0 blir mängdenasymptotiskt likafördelad i M med avseende på det hyperboliska

volymsmåttet, så länge vi håller βi − αi ≥ y1/2−ε för alla 1 ≤ i ≤ n.I Artikel II studerar vi fördelningen av längder av nollskilda gitter-

vektorer i ett slumpvis valt gitter av hög dimension. Givet ett gitterL ∈ Xn, ordnar vi dess nollskilda vektorer efter växande längd som±v1,±v2,±v3, . . . och låter `j = |vj | (alltså gäller 0 < `1 ≤ `2 ≤ `3 ≤. . .). Dessutom definierar vi

Vj =πn/2

Γ(n2 + 1)`nj ,

19

så att Vj betecknar volymen av en n-dimensionell boll med radie `j .Inspirerade av resultat av Rogers och Schmidt visar vi att då n → ∞konvergerar följden Vj∞j=1 i fördelning mot punkterna i en Poissonpro-cess på R+ med konstant intensitet 1

2 .I Artikel III fortsätter vi att studera statistiska egenskaper hos följden±v1,±v2,±v3, . . . av nollskilda vektorer i ett slumpvis valt gitter avhög dimension. Bland annat visar vi att vinklarna mellan de kortastenollskilda vektorerna i ett slumpvis valt gitter L ∈ Xn konvergerar ifördelning mot en familj av oberoende Gaussiska variabler då n→∞.I Artikel IV och Artikel V studerar vi den asymptotiska värdefördelnin-

gen hos Epsteins zeta funktion som för L ∈ Xn och Re s > n2 definieras

genom

En(L, s) =∑m∈L

′|m|−2s,

där ′ betecknar att vi endast summerar över nollskilda gittervektorer.Huvudresultatet i Artikel IV säger att om vi fixerar c > 1

2 så kan denasymptotiska värdefördelningen då n → ∞ hos En(·, cn) (lämpligt nor-maliserad) fullständigt beskrivas i termer av en Poissonprocess på R+

med konstant intensitet 12 .

I Artikel V studerar vi den asymptotiska värdefördelningen hosEn(L, s) för 0 < Re s < n

2 och ett slumpvis valt gitter L ∈ Xn. Giveten Poissonprocess P =

N(V ), V ≥ 0

på R+ med konstant intensitet

12 låter vi T1, T2, T3, . . . beteckna processens punkter ordnade så att0 < T1 < T2 < T3 < . . .. Vi sätter N(V ) = 2N(V ) och definierar, föralla V ≥ 0,

R(V ) = N(V )− V.

Dessutom låter vi Vn beteckna volymen av enhetsbollen i Rn. Ett avhuvudresultaten i Artikel V säger att om vi för 1

4 < c1 < c2 <12 betraktar

c 7→ V −2cn En(·, cn)

som en stokastisk variabel med värden i C([c1, c2]

)så konvergerar den i

fördelning mot

c 7→∫ ∞

0

V −2c dR(V )

då n→∞. Detta resultat tillämpas sedan på flera problem av statistisknatur på rummet Xn. En viktig del av beviset av vårt huvudresultatär en uppskattning som visar sig vara av intresse även i samband medGauss cirkelproblem generaliserat till godtycklig dimension och generellaellipsoider.

20

Acknowledgements

First of all I would like to express my deep gratitude to my advisorAndreas Strömbergsson for invaluable help; for constant support andencouragement; for providing me with interesting research problems; forall interesting discussions and long emails on mathematics; for alwaysreading my manuscripts very carefully and giving valuable comments;and for giving my kayak a place to live.I am grateful to my second advisor Dennis Hejhal for good advice, for

many interesting discussions (also on other subjects than mathematics)and for sharing his enormous enthusiasm and expertise.I am also grateful to Svante Janson for reading several of my

manuscripts and giving valuable comments and suggestions.During my years as a student in Uppsala I have had the privilege to

learn mathematics from many great teachers. In particular I would like tomention Anders Vretblad, Dennis Hejhal, Karl-Heinz Fieseler, RyszardRubinsztein and Burglind Jöricke.Furthermore I would like to thank my fellow PhD-students and col-

leagues at the Department of Mathematics for creating a nice workingatmosphere. A special thanks to my former room mate Johan Björklundfor endless discussions and problem sessions at our gigantic whiteboard.I also thank my family and friends for keeping my mind off research

when needed, and for not asking too many questions about what I amreally working on.Finally, I thank Tina, for her never ending support and encouragement,

and for her ability to cope with me even though I quite often spend toomuch time at work.

21

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