1. prime numbers and prime orbits 2. sums of squares and closed
TRANSCRIPT
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
1 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
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
2 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
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.
3 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
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 ...
4 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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, · · · .
5 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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 ...
6 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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”
7 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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
8 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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.
10 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Prime Number TheoremRiemann zeta functionPrime geodesic theoremDynamical zeta function
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.
11 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
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
12 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Sums of squaresClosed geodesics in homology classes
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.
13 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Sums of squaresClosed geodesics in homology classes
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.
14 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Sums of squaresClosed geodesics in homology classes
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 .
15 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Sums of squaresClosed geodesics in homology classes
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 → +∞.
16 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
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?
17 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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
18 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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 → +∞.
19 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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).
20 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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 ...
21 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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
).
22 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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.
23 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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 → +∞.
24 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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 → +∞.
25 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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
26 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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.
27 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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.
28 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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?
29 / 30
Introduction1. Prime Numbers and Prime Orbits
2. Sums of squares and closed geodesics in homology3. Circle problem and hyperbolic circle problem
Circle problemHyperbolic circle problemSketch of proofApollonian circle packings
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.
30 / 30