Riemann zeta function
Selberg zeta function
More general zeta functions
Dynamical zeta functions - Lecture 2
Mark Pollicott
June 11, 2010
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Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
Different types of zeta functions
Special values
and volumes
Number Theory
Riemann ζ−function
Geometry
Mahler measures
Class numbers of
binary quadratic forms
Dynamical Systems
QUE
Selberg ζ−function
Ruelle ζ−function
Surfaces with κ =−1
Surfaces with κ <0
"Riemann Hypothesis"
2 / 24
Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
The original and best: The Riemann zeta function
The Riemann zeta function is the complex function
ζ(s) =
∞
X
1
s
n
n=1
which converges for Re(s) > 1. It is convenient to write this as an Euler product
ζ(s) =
Y`
´−1
1 − p −s
p
where the product is over all primes p = 2, 3, 5, 7, 11, · · · .
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Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
Aside on hyperbolic tetrahedra
Consider the three dimensional hyperbolic space
H3 = {x + iy + jt : z = x + iy ∈ C, t ∈ R+ }
and the Poincaré metric
ds 2 =
dx 2 + dy 2 + dt 2
t2
8
i
Given z ∈ C consider the hyperbolic
tetrahedron ∆(z) with vertices
{0, 1, z, ∞} for some z ∈ C.
Let D(z) denote the volume of ∆(z)
0
1
z
Lemma (Lobachevsky, Milnor)
If three faces meeting at a vertex have anglesRα, β, δ between them, then
D(z) = L(α) + L(β) + L(δ) where L(α) = − 0α log |2 sin(t)|dt, etc.
These play an interesting role in, for example, Gromov’s proof of the Mostow rigidity
theorem.
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Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
1. Special values: Zagier’s theorem
P
π2
1
It is easy to see that ζ(2) = ∞
n=1 n2 = 6 .
More generally, the Dedekind zeta function ζK (s) is the analogue of ζ(s) for
extensions K ⊃ Q.
Theorem (Zagier)
The Dedekind zeta function ζK (s) has values ζK (2) which are related to volumes of
hyperbolic tetrahedra.
Example
√
When K = Q( −7) then the Dedekind zeta function takes the form
ζQ(√−7) (s) =
and
ζQ(√−7) (2) =
4π 2
√
21 7
1
2
X
(x,y )6=(0,0)
„
„
2D
1+
√
2
1
(x 2 + 2xy + 2y 2 )s
−7
«
„
+D
√
««
−1 + −7
2
(Numerically, the last expression is approximately 1 · 1519254705 · · · )
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Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
2. Zeros: Riemann hypothesis
The following conjecture was formulated by Riemann in 1859 (repeated as Hilbert’s
8th problem).
Riemann Hypothesis
The non-trivial zeros lie on Re(s) = 12 .
Why should we care? This has very implications for counting the number of primes
≤ x. Recall that:
Theorem (Prime number theorem)
The number π(x) of primes numbers less than x satisfies
π(x) ∼
x
as x → +∞.
log x
The Riemann hypothesis would improve this to
Z x
“
”
1
π(x) =
du + O x 1/2 log x
2 log u
R
(where 2x log1 u du ∼ logx x )
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Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
Some evidence: A partial result on the location of the zeros
G. H. Hardy (1887-1947)
Hardy’s Theorem
Hardy showed that 7/10 of the zeros lie on the line Re(s) = 21 .
The Riemann Conjecture topped G.H.Hardy famous wish list from the 1920s:
1
2
3
4
5
6
Prove the Riemann Hypothesis.
Make 211 not out in the fourth innings of the last test match at the Oval.
Find an argument for the nonexistence of God which shall convince the general
public.
Be the first man at the top of Mt. Everest
Be proclaimed the first president of the U.S.S.R., Great Britain and Germany.
Murder Mussolini.
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Riemann zeta function
Selberg zeta function
More general zeta functions
Back to Number Theory
1. Special values
2. Zeros
The Hilbert-Polya approach to the Riemann hypothesis
D. Hilbert (1862-1943)
G. Polya (1887-1985)
Hilbert and Polya are associated with the idea of tying to understand the location of
the zeros in terms of eigenvalues of some (as of yet) undicovered self-adjoint operator
(which necessarily has real eigenvalues somehow related to the zeros).
This idea has yet to reach fruition for the Riemann zeta function (despite interesting
work of Berry-Keating, Connes, etc.) but the approach works particularly well for the
Selberg Zeta function ...
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Riemann zeta function
Selberg zeta function
More general zeta functions
The Definition
1. Zeros
2. Special values
The Selberg Zeta function
Recall that the Selberg zeta function for a compact surface V of negative curvature
κ = −1 is formally defined by
Z (s) =
∞ Y“
”
Y
1 − e −(s+n)l(γ)
n=0 γ
where s ∈ C and γ denotes a closed geodesic of length l(γ).
Theorem
Z (s) converges for Re(s) > 1 and extends analytically to C.
If we denote ζV (s) = Z (s + 1)/Z (s) then we can write
ζV (s) =
Y“
1 − e −sl(γ)
”−1
γ
which is perhaps closer in appearance to the Riemann zeta function
Y`
´−1
ζ(s) =
1 − p −s
γ
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Riemann zeta function
Selberg zeta function
More general zeta functions
The Definition
1. Zeros
2. Special values
1. Zeros of the Selberg Zeta function
Let ∆ : L2 (V ) → L2 (V ) be the Laplace operator given by
∆ = y2
„
∂2
∂2
+
2
∂x
∂y 2
«
This is a self-adjoint operator and we are interested in the solutions
0 = λ0 ≤ λ1 ≤ λ2 ≤ · · ·
for the eigenvalue equation ∆φn = −λφn .
The location of the zeros sn for Z (s) (i.e., Z (sn ) = 0) are described in terms of the
eigenvalues λn for the Laplacian
Theorem
The zeros of the Selberg zeta function Z (s) can be described by:
1
2
3
s = 1 is a zero.
q
sn = 12 + i 14 − λn , for n ≥ 1, are “spectral zeros”.
s = −m, for m = 0, 1, 2, · · · , are “trivial zeros”.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The Definition
1. Zeros
2. Special values
Spectral zeros of the Selberg Zeta function
The spectral zeros lie on the “cross” [0, 1] ∪ [ 12 − i∞,
1
2
+ i∞]:
Re(s) = 1/2
0
1
Remark
The proof of these results is usually based on using the Selberg trace formula to
extend the logarithmic derivative Z 0 (s)/Z (s) (cf. Hejhal, SLN 548).
Remark
If V is non-compact, but of finite area (such as the modular surface) then there are
extra zeros which correspond to those for the Riemann zeta function.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The Definition
1. Zeros
2. Special values
2. Special values: Determinant of the Laplacian
We would like to find a way to define a determinant “det(∆) =
Q
n
λn ”.
Lemma (Seeley)
The Minakshisundaram-Pleijel zeta function
η(s) =
∞
X
λ−s
n
n=1
has an analytic extension to the complex plane.
We define the Determinant of the Laplacian by:
`
´
det(∆) = exp −η 0 (0)
This has a connection with the Selberg zeta function:
Theorem
There exists C > 0 (depending only on the genus of the surface) such that
det(∆) = CZ 0 (1).
Sarnak, Wolpert and others have studied the dependence of det(∆) on the choice of
metric on the surface V . (But there are still questions about, say, the critical points)
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Riemann zeta function
Selberg zeta function
More general zeta functions
The Definition
1. Zeros
2. Special values
2. Other special values: Resolvents and QUE (from A to Z)
Definition
R
R
Let A ∈ C ω (SV ) with Ad(vol) = 0. For each closed geodesic γ let A(l) = γ A
denote the integral of A around γ. We then formally define, say,
d(s, z) =
∞ Y“
Y
1 − e zA(γ)−(s+n)l(γ)
”
for s, z ∈ C
n=0 γ
Theorem (Anarathaman-Zelditch)
The function d(s, z) is analytic for s, z ∈ C. The logarithmic derivative
1
∂
η(s) := d(s,0)
d(z, s)|z=0 has poles at s = sn .
∂z
Moreover, the residues res(η, sn ) are related to the eigenfunctions for −∆ giving:
Question (Quantum Unique Ergodicity: Rudnick-Sarnak)
Does res(η, sn ) → 0 as n → +∞?
cf. Shnirelman, Colin de Verdiere, Zelditch,N. Anantharaman, E. Lindenstrauss
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Recall: Dynamical approach to the Selberg Zeta function
Basic Approach
We associate to an analytic expanding (interval) map T : I → I (e.g., the Gauss map
in the case of the continued fraction transformation) to the geodesic flow.
We can then rewrite the Selberg zeta function as
Z (s) = 1 +
∞
X
an (s)
n=1
where
1
an (s) only depends on periodic points T of period ≤ 2n;
2
There exists c > 0 such that for each s there exists C > 0 such that
2
|an (s)| ≤ Ce −cn . In particular, the series converges for all s ∈ C.
Corollary
We have an expression for the Selberg zeta function
Z (s) = 1 +
∞
X
an (s)
n=1
which converges for all s ∈ C.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
More general transformations and zeta functions
Of course, given any suitable
T we can also associate a zeta function.
‘
For example, let X = ∞
i=1 be a disjoint union of n-dimensional simplices Xi and let
T : X → X be maps which:
1
Expand distances locally; and
2
T |Xi is analytic; and
3
Markov (i.e., each image T (Xi ) is a union of some of the other simplices)
We might then generalize the definition of the zeta function for the continued fraction
transformation to:
!
∞
X
X
1
| det(DT n )(x)|−s
Z (s) = exp −
Zn (s) where Zn (s) =
n
det(I
− (DT n )0 (x)−1 )
n=1
x∈Fix(T n )
and the same methods give that Z (s) has an extension to C.
Simple Philosophy
Any quantities which can be expressed in terms of the (dynamical) zeta functions can
often be easily computed numerically.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Application: Computing Hyperbolic Volumes
Recalling that volumes of hyperbolic tetrahedra appeared in Zagier’s work:
Corollary
The largest possible volume of a hyperbolic tetrahedron in H3 is
1.01494160640965362502 . . .
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Computing other quantities
Other examples of eminently computable quantities are:
1
Integrals and the Mahler measures of polynomials
2
The dimension of the limit set of Schottky groups (or, equivalently, smallest
eigenvalues of the Laplacian).
3
Lyapunov exponents for random products of matrices.
4
The determinant of the laplacian det(∆).
5
Regularity of wavelets.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Application: Integrals and Mahler measures
More generally, the estimates on zeta functions can be used to integrate analytic
functions f : [0, 1] → R (with respect to Lebesgue measure)
Theorem (Jenkinson-P)
We can approximate the integral
1
Z
2
f (x)dx = mn (f ) + O(e −(log 2−)n )
0
where mn (f ) is explicitly given in terms of the values at f at the 2n periodic points for
the doubling map Tx = 2x (mod 1).
In particular, if f (x) is a polynomial then one defines the Mahler measure by:
„
M(f ) = exp
1
2π
Z
2π
«
log (|f (e iθ )|)dθ .
0
In particular, generalizations of these often come up in formulae the entropy of Zd
actions.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Application: Dimension of Limit sets
Let Γ ⊂ PSL(2, C) be a non-elementary Schottky Kleinian group generated, for
example, by reflection in a finite number of circles in C with disjoint interiors. Let Λ
be the associated limit set .
Theorem (Bowen)
The Hausdorff dimension of the limit set dim(Λ) is a zero of the associated zeta
function Z (s)
We can associate and expanding dynamical system T : Λ → Λ.
Theorem (Schottky groups)
For each N ≥ 1 we can explicitly define a sequence of values sN depending on the
periodic points for T of period n, for 1 ≤ n ≤ N, and associate C > 0 and 0 < δ < 1
such that
3/2
|dim(Λ) − sN | ≤ C δ N .
The sN are zeros of approximations to the determinant using only the derivatives
Dz T n evaluated at period-n points z, for 1 ≤ n ≤ N.
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
An example of McMullen
Consider three circles C0 , C1 , C2 ⊂ C of equal radius, arranged symmetrically around
the unit circle S 1 , each intersecting S 1 orthogonally, and meeting S 1 in an arc of
length θ, where 0 < θ < 2π/3. Let Λθ ⊂ S1 denote the limit set associated to the
group Γθ of transformations given by reflection in these circles.
C1
θ
1
S
θ
1
C2
C0
θ
For example, with the value θ = π/6 we show that the dimension of the limit set Λπ/6
is
dim(Λπ/6 ) = 0 · 18398306124833918694118127344474173288 . . .
which is accurate to the 38 decimal places given.
(Equivalently, the smallest eigenvalue of the laplacian on H3 /Γ is
λ0 = dim(Λπ/6 )(2 − dim(Λπ/6 )) = 0 · 3341163556703682452613106798303932895)
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Application: Lyapunov exponents for Random matrix products
1
Let A1 , A2 ∈ GL(2, R) be 2 × 2 non-singular matrices with real entries.
2
Let k · k be a norm on matrices (e.g., kAk2 = trace(AAT )).
Definition
The (largest) Lyapunov Exponent is
λ := lim
n→∞
1 1
2n n
|
X
log kAi1 · · · Ain k
i1 ,··· ,in ∈{1,2}
{z
=:λn
}
Question
How do we estimate λ?
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Lyapunov exponents for Random matrix products
For each n ≥ 1, we can again consider the matrices Ai1 · · · Ain , where
i1 , · · · , in ∈ {1, 2}.
Theorem
For each n ≥ 1 one can find approximations ξn which converge
to
“
”λ
2
superexponentially ( i.e., ∃β > 0 such that |λ − ξn | = O e −βn ).
The numbers ξn are explicitly defined in terms of the entries and eigenvalues of these
2n matrices.
The number of matrices we need to compute ξn still grows exponentially, i.e.,
]{Ai1 · · · Ain : i1 , · · · , in ∈ {1, 2}} = 2n .
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Riemann zeta function
Selberg zeta function
More general zeta functions
The dynamical approach to the Selberg Zeta function
Dynamical approach to more general functions
Application: Computing numerical values
Example: Lyapunov exponents for Random matrix products
Example
Let p1 = p2 =
1
2
and choose
„
A1 =
2
1
1
1
«
„
and A2 =
3
2
1
1
«
then with n = 9:
`
´
λ = 1 · 1433110351029492458432518536555882994025 +O 10−40
|
{z
}
=ξn=9
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