1. Prime Numbers and Prime Orbits 2. Sums of squares and closed

Introduction1. Prime Numbers and Prime Orbits

2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem

Dynamical zeta functions and counting

Mark Pollicott

June 10, 2010

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Three results

We recall three asymptotic results in number theory

1 The prime number theorem

2 The sums of squares

3 The circle problem

Aim

To describe geometric “analogues” of these results, for which there are dynamical orergodic theoretic proofs.

1 The prime orbit theorem

2 The closed geodesics null in homology

3 The hyperbolic circle problem for orbit counting

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Something I hope to talk about (at the end)

The following is the January 2009 page from the free calender on the ”Theorem of theday” webpage: http://www.theoremoftheday.org/

A Theorem on Apollonian Circle Packings For every integral Apollonian circle packing there is aunique ‘minimal’ quadruple of integer curvatures, (a, b, c, d), satisfying a ≤ 0 ≤ b ≤ c ≤ d, a+b+c+d > 0and a + b + c ≥ d. This so-called root quadruple completely specifies the packing.

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A Descartes configuration consists of four mutually tangent circles. Above right, for example, is a circle of radius 1/7 containing circles ofradius 1/12, 1/17 and 1/20, each of which has a point of contact with the other three. The integers labelling the circles are the curvatures(the reciprocals of the radii) and in the root quadruple of curvatures, (−7, 12, 17, 20), the enclosing circle of radius 1/7 is determined to havenegative curvature so that all four circles have disjoint interiors. Any such configuration specifies four more tangent circles — above right, thesehave curvatures 24, 33, 48, and 105, producing four new configurations (−7, 12, 17, 24), (−7, 12, 20, 33), (−7, 17, 20, 48) and (12, 17, 20, 105).Repeating this process produces a system of infinitely packed circles: an Apollonian circle packing. If our initial configuration is integral, as ineach of the above examples (which are drawn to different scales), then we will get an integral packing with every curvature an integer.

This theorem comes from a series of four pivotal papers by the AT&T team of Ronald Graham, Jeffrey Lagarias, Colin Mallowsand Allan Wilks, together with Catherine Yan of Texas A&M University. They further show that all integral Apollonian circlepackings may be derived from root quadruples which, like those depicted above, have entries whose gcd is 1.Web link: www.ams.org/featurecolumn/archive/kissing.html. The packing images were provided by Emil Vaughan.Further reading: Introduction to Circle Packing: The Theory of Discrete Analytic Functions by Kenneth Stephenson, CUP, 2005.

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Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function

1. Prime Number Theorem and Prime Orbit Theorem

The following classic theorem was proved independently by Hadamard and de la ValleePoussin.

Theorem (Prime number theorem)

The number π(x) of primes numbers less than x satisfies

π(x) ∼x

log xas x → +∞.

i.e., limx→+∞π(x)

x/ log x = 1.

The original proof uses the Riemann zeta function ...

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The original and best zeta function: The Riemann zeta function

The Riemann zeta function is the complex function

ζ(s) =∞X

n=1

1

ns

which converges for Re(s) > 1. It is convenient to write this as an Euler product

ζ(s) =Y

p

`1− p−s´−1

where the product is over all primes p = 2, 3, 5, 7, 11, · · · .

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Basic properties of the Riemann zeta function

Key properties of ζ(s)

To show the Prime Number Theorem it is enough to know:

1 ζ(s) has a simple pole at s = 1;

2 ζ(s) otherwise has a non-zero analytic extension to a neighbourhood of Re(s) = 1

One can then use a Tauberian theorem or the residue theorem to convert this into theasymptotic result in the Prime Number Theorem.

Simple Philosophy

The more we know about the domain of the zeta function ζV (s) the better theasymptotic approximation we can get for the number of primes π(x) less than x .

In particular, if we knew ...

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Aside: Riemann Hypothesis

The following conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s8th problem).

Riemann Hypothesis

The non-trivial zeros lie on Re(s) = 12 .

Knowing the Riemann hypothesis would improve the Prime Number Theorem to:

Conjecture

π(x) =

Z x

2

1

log udu + O

“x1/2 log x

”

(whereR x2

1log u du ∼ x

log x )

In the absence of the Riemann Conjecture the orginal proof of Hadamard and de laVallee Poussin shows that there exists C > 0 with no zeros in a region

{s = σ + it : σ > 1− C/ log |t| and |t| ≥ 1}

which is still enough to show there exists a > 0

π(x) =

Z x

2

1

log udu + O

“xe−a

√log x”

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A dynamical analogue: Geodesic flows

Example (Geodesic flow)

Let V be a compact surface with curvature κ < 0, for x ∈ V . Let

M = SV := {(x , v) ∈ TM : ‖v‖x = 1}

then we let φt : M → M be the geodesic flow, i.e., φt(v) = γ(t) where γ : R → V isthe unit speed geodesic with γ(0) = (x , v).

x

v

x

v

γv

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Prime geodesic theorem

Question

What is the geometric analogue of the Prime Number Theorem?

Denote by γ closed geodesics of length l(γ) (which correspond o closed orbits for thegeodesic flow).

Definition

Let N(x) denote the number of closed geodesics with ehl(γ) ≤ x

Theorem (Prime geodesic theorem)

There exists h > 0 such that satisfies

N(x) ∼x

log xas x → +∞.

Equivalently, writing π(T ) := Card{γ : l(γ) ≤ T} we have

π(T ) ∼ehT

hTas T → +∞.

The proof uses a dynamical zeta function.9 / 30




Definition of the dynamical Zeta function

We can define a zeta function by

ζV (s) =Y

γ

“1− e−sl(γ)

”−1

by analogy with the Riemann zeta function ζ(s) =Q

γ

`1− p−s

´−1, i.e.

primes p closed geodesics γp ≤ x el(γ) ≤ x

Theorem

There exists h > 0 such that:

1 ζV (s) converges for Re(s) > h;

2 ζV (s) has a simple pole at s = h;

3 ζV (s) otherwise has a non-zero analytic extension to a neighbourhood ofRe(s) = h

The same proof as for the Prime Number Theorem now gives the Prime OrbitTheorem.

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Aside: The analogue of the Riemann Hypothesis

A partial partial analogue of the Riemann Hypothesis holds for surfaces with κ < 0.

Lemma

There exists ε > 0 such that ζV (s) has no more zeros in Re(s) > 1− ε.

In particular ...

Theorem

There exists ε > 0 such that

N(x) =

Z x

2

1

log udu + O

`x1−ε´

For surfaces with constant curvature κ = −1 this follows from the Selberg traceformula. For surfaces with variable curvature κ < 0 this follows from work ofDolgopyat on exponential mixing for geodesic flows.

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Sums of squaresClosed geodesics in homology classes

2. Sums of squares

Consider the natural numbers which sums of two squares 1, 2, 4, 5, ..., u2 + v2, · · ·(where u, v ∈ Z+). For example,

1 = 02 + 12

2 = 12 + 12

4 = 02 + 22

5 = 12 + 22

8 = 22 + 22

...

Definition

Let S(x) = Card{u2 + v2 ≤ x} be the number of such sums of squares less than x .

Theorem (Landau, Ramanujan)

There exists b > 0 such that

S(x) ∼ bx

√log x

as x → +∞,

i.e., limx→+∞ S(x)/( x√log x

) = b

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Landau and Ramanujan

The theorem was originally proved by Landau in 1908.

This theorem was independently stated in Ramanujan’s first famous letter toHardy from 16 January 1913.

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Landau’s proof

In place of the zeta function, one uses another complex function

η(s) =∞X

n=1

bnn−s where bn =

(1 if n = u2 + v2 is a sum of squares

0 if n is not a sum of squares

This converges for Re(s) > 1. But this has an algebraic pole at s = 1:

Lemma

We can write

η(s) =C

√s − 1

+ A(s)

where A(s) is analytic in a neighbourhood of Re(s) ≥ 1.

Using complex analysis one can write

M(x) =1

2π

Z c+i∞

c−i∞η(s)

xs

sds for c > 1

and the asymptotic formula comes from moving the line of integration to the left.

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A geometric analogue: Closed geodesics in homology classes

In counting geodesics, we can count the number π0(T ) of closed geodesics whoselength is at most T and which are null in homology.

Theorem

For a surface of negative curvature and genus g > 1 there exists c0 such that

π0(T ) ∼ c0ehT

Tg+1as T → +∞

We can compare this with counting closed geodesics with no restrictions gave

π(T ) ∼ ehT

hT .

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Remarks

For both problems one can actually get more detailed asymptotic formulae.

Theorem (Expansion for sums of squares)

There exist constants b0, b1, b2 · · · such that

N0(T ) =T

√log T

(b0 +b1

log T+

b2

(log T )2+ · · · )

as T → +∞.

Theorem (Expansion for null closed geodesics )

For a compact surface of (variable) curvature κ < 0 and genus g > 1 there existconstants c0, c1, c2 · · · such that

π0(T ) =ehT

Tg+1(c0 +

c1

T+

c2

T 2+ · · · )

as T → +∞.

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Circle problemHyperbolic circle problemSketch of proofApollonian circle packings

3. Circle problem

Finally, we come to the third type of asymptotic result:

Definition

Let N(r) = {(m, n) : m2 + n2 ≤ r2} denote the number of pairs (n, m) ∈ Z2 at adistance at most r from the origin.

It is easy to see that N(r) ∼ πr2.

Gauss showed that N(r) = πr2 + O(r).

Conjecture

For each ε > 0 we have that N(r) = πr2 + O(r1/2+ε)

Question

What happens if we replace R2 by the Poincare disk D2, and we replace Z2 by adiscrete (Fuchsian) group of isometries?

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Hyperbolic Circle problem: cocompact Fuchsian groups

Let D2 = {z ∈ C : |z| < 1} be the unit disk with the Poincare metric

ds2 =dx2 + dy2

(1− (x2 + y2))2.

of constant curvature κ = −1.

Consider the action g : D2 → D2 defined by g(z) = az+bbz+a

for g lying in a discrete group

Γ ⊂„

a bb a

«: a, b ∈ C, |a|2 − |b|2 = 1

ff

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Hyperbolic circle problem: Compact surfaces

Let Γ0 denote the orbit of the fixed reference point 0 ∈ D2.

Question

What are the asymptotic estimates for the counting function

N(T ) = Card{g ∈ Γ : d(0, g0) ≤ T}

as T → +∞?

T

0

Theorem

Assume that Γ is cocompact, i.e., the quotient surface V = D2/Γ is compact. Thenthere exists C > 0 such that N(T ) ∼ CeT as T → +∞.

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Hyperbolic Circle problem: Schottky groups

Assume that Γ is a Schottky group (in particular, Γ is a free group)

T

0

Theorem

There exists C > 0 and δ > 0 such that

N(T ) := Card{g ∈ Γ : d(0, g0) ≤ T} ∼ CeδT

as T → +∞.

Here 0 < δ < 2 is the Hausdorff dimension of the limit set (i.e., the accumulationpoints of Γ0 on the unit circle).

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1st step of proof: Poincare series

In place of the zeta function one now studys the following complex function:

Definition

We define the Poincare series by

η(s) =X

g∈Γ

e−sd(g0,0).

This converges to an analytic function for Re(s) > δ.

Lemma

The Poincare series has a meromorphic extension to C with:

a simple pole at s = 1; and

no poles other poles on the line Re(s) = δ.

When D2/Γ is compact then this result is based on the Selberg trace formula.However, when Γ is Schottky group a dynamical approach works better ...

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2nd step in the proof: Poincare series and shift spaces

Consider the model case of a Schottky group Γ = 〈a, b〉.Each g ∈ Γ− {e} can be written g = g1g2 · · · gn, say, where gi ∈ {a, b, a−1, b−1}with gi -= g−1

i+1.

Definition

Consider the space of infinite sequences

ΣA = {(xn)∞n=0 : A(xn, xn+1) = 1 for n ≥ 0}, where A =

„1 1 0 11 1 1 00 1 1 11 0 1 1

«

and the shift map σ : ΣA → ΣA given by (σx)n = xn+1.

We then have the following way to enumerate the displacements d(g0, 0)

Lemma

There exists a (Holder) continuous function f : ΣA → R and x ′ ∈ ΣA such that thereis a correspondence between the sequences {d(g0, 0) : g ∈ Γ} and

(n−1X

k=0

f (σky) : σny = x ′, n ≥ 0

).

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3rd step in the proof: Extending the Poincare series

Corollary

We can write that

η(s) =∞X

n=1

X

σny=x0

exp

−s

n−1X

k=0

f (σky)

!

Moreover, if we define families of (transfer) operators Ls : C(ΣA) → C(ΣA), s ∈ C, by

Lsw(x) =X

σy=x

e−sf (y)w(y), where w ∈ C(ΣA)

then we can formally write

η(s) =∞X

n=1

Lns 1(x).

Finally, the better spectral properties of the operators Ls on the smaller space ofHolder have the meromorphic extension and the other properties.

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4. Asymptotic formulae

Let M(T ) = Card{g ∈ Γ : exp(hd(x , gx)) ≤ eT } then we can write

η(s) =

Z ∞

1t−sdM(t)

and employ the following standard Tauberian Theorem

Lemma (Ikehara-Wiener)

If there exists C > 0 such that

ψ(s) =

Z ∞

1t−sdM(t)−

C

s − δ

is analytic in a neighbourhood of Re(s) ≥ h then M(T ) ∼ CT as T → +∞.

We can deduce from M(T ) ∼ CT that

N(T ) := Card{g ∈ Γ : d(0, g0) ≤ T} ∼ CehT

as T → +∞.

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Generalizations to variable curvature

More generally, assume that:

V is a surface of (variable) negative curvature;

eV is the universal covering space for V with the lifted metric d ;

The covering group Γ = π1(V ) acts by isometries on eV and V = eV /Γ.

We want to find asymptotic estimates for the counting function

N(T ) = Card{g ∈ Γ : d(x , gx) ≤ T}.

Let h > 0 be the topological entropy.

Theorem

There exists C > 0 such that N(R) ∼ CehT as T → +∞.

Conjecture

There exists C > 0 and ε > 0. N(T ) = CehT + O(e(h−ε)T ) as T → +∞.

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Apollonian circle packings

Consider a circle packing where four circles are arranged so that each is tangent to theother three.

Assume that the circles have radii r1, r2, r3, r4 and denote their curvatures by ci = 1ri.

Lemma (Descartes)

The curvatures satisfy 2(c21 + c2

2 + c23 + c2

4 ) = (c1 + c2 + a3 + c4)2

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Apollonian circle packings: Best known example

Consider the special case c1 = c2 = 0 and c3 = c4 = 1.

The curvatures of the circles are 0, 1, 4, 9, 12, · · ·

Question

How do these numbers grow?

We want to count these curvatures with their multiplicity.

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Asymptotics on curvatures

Definition

Let C(T ) be the number of circles whose curvatures are at most T .

There is the following asymprotic formula.

Theorem (Kontorovich-Oh)

There exists K > 0 and δ > 0 such that

C(T ) ∼ KT δ

as T → +∞.

Remarks

The value δ = 1 · 30568 · · · is the dimension of the limit set.

The value K = 0 · 0458 · · · can be estimated too

The proof is dynamical and comes from reformulating the problem in terms of adiscrete Kleinian group acting on three dimensional hyperbolic space.

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Idea of proof

Given a family of tangent circles (in black) leading to a circle packing, weassociate a new family of tangent circles (red).

One then defines isometries of three dimensional hyperbolic space H3 byreflecting in the associated geodesic planes (= hemispheres in upper half-space).

Let Γ be the Schottky group (for H3) then one can reduce the theorem to acounting problem for Γ.

Question

Can these results be generalized to other circle packings?

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Things I wasn’t able/capable to talk about

Of course there have been many important contributions to number theory usingergodic theory:

The first of Khinchin’s pearls was van de Waerden’s theorem (on arithmeticprogressions). This, and many deep generalizations, have ergodic theoretic proofsby Furstenberg, etc.

Margulis’ proof of the Oppenheim Conjecture (on the values of indefinitequadratic forms)

Einseidler-Katok-Lindenstrauss work on the Littlewood conjecture (onsimultaneous diophantine approximations).

Benoist-Quint work on orbits of groups on homogeneous spaces.

However, this lecture isn’t concerned with these - and we return to our theme of zetafunctions, and similar functions.

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Documents

1. Prime Numbers and Prime Orbits 2. Sums of squares and closed