dynamical and spectral zeta functions
Since the discovery of Riemann's zeta function and its
role in the distribution of prime numbers, many other 'zeta functions'
have been defined in analogy with Riemann's. Although there is, as yet, no clear definition
as to what precisely constitutes a zeta function, the general feeling amongst mathematicians
and physicists is that "we know one when we see one".
This page concerns zeta functions which are associated with dynamical systems,
or with any physical system possessing a spectrum of eigenvalues. The fact that these
zeta functions are analogous in some way to Riemann's ("the grandmother of all zeta functions")
leads to some interesting correspondences between number theoretical and physical phenomena.
Chris Hillman, in
a 23/10/01 posting
to sci.physics.research explains:
"One of the reasons why dynamical zeta functions and the associated
transfer operators are so important in dynamical systems is that if
you know enough about the location of the zeros/poles, you can estimate
the rate of decay of correlations, i.e. the mixing ("randomization")
rate; see for example the book
V. Baladi, Positive Transfer Operators and Decay of Correlations,
Advanced Series in Nonlinear Dynamics, Vol. 16. World Scientific, 2000
Note that these ideas relate number theory, exactly solvable models
(YangBaxter, etc.), and symbolic dynamics."
[Note that the same post included a claim that the Riemann zeta function
acts as the dynamical zeta function for a particular symbolic dynamical
system (a shift), which turns
out to be misguided.]
According to Kitchen's book on symbolic dynamics, the dynamical zeta function was first
introduced in:
M. Artin and B. Mazur, "On periodic points", Annals of Mathematics 81 (1965) 8299.
In this article, Artin and Mazur define the dynamical zeta function initially in a form which "coincides with the zeta function introduced by Lang":
S. Lang, "Lseries of a covering", Proceedings of the National Academy of Sciences 42 No. 7 (1956) 422424
From http://www.mathphyszeta.de/arithmetische.html [site now defunct]:
"The arithmetical theory of zeta functions:
The dynamical theory of zeta functions is based on analogies with the theory of arithmetical zeta functions,
especially those associated with varieties over finite fields. The first really explicit connection was made by
B. Mazur and M. Artin in the early 1960's. This connection has enriched both areas of research.
On the one hand analytic number theory offers a wide range of techniques and insights which are applicable in
dynamical theories. In particular these are of considerable use in applying the results derived about zeta functions
to specific problems.
In the other direction the dynamical theory gives a framework in which one can think about some of the deeper problems
of the theory of arithmetical zeta functions. Although these are very intriguing they remain very much at the level of speculation."
D. Ruelle, "Dynamical zeta functions and transfer operators", IHES
report IHES/M/02/66 (August, 2002)  a very nice survey article (in PDF format)
[excerpt from introduction:] "The simplest invariant measures for a dynamical system are those carried by periodic orbits. Counting periodic orbits is thus a natural task from the point of view of ergodic theory. And dynamical zeta functions are an effective tool to do the counting. The tool turns out to be so effective in fact as to make one suspect that there is more to the story than what we currently understand."
D. Ruelle, "Dynamical zeta functions:
Where do they come from and what are they good for?" from Mathematical
Physics X, ed. K. Schmudgen (Springer, 1991)
D. Ruelle, "Dynamical
zeta functions for maps of the interval [0,1]", Bulletin of the AMS (New Series)
30 (1994) 212214.
[abstract:] "A dynamical zeta function $\zeta$ and a transfer operator $\scr L$ are
associated with a piecewise monotone map f of the interval [0,1] and a weight function
g. The analytic properties of $\zeta$ and the spectral properties of $\scr L$ are related
by a theorem of Baladi and Keller under an assumption of 'generating partition'. It is
shown here how to remove this assumption and, in particular, extend the theorem of Baladi
and Keller to the case when f has negative Schwarzian derivative."
WWN notes on dynamical zeta functions
(part of a workinprogress)
J. Lagarias, "Number theory zeta functions and dynamical zeta functions",
in Spectral Problems in Geometry and Arithmetic (T. Branson, ed.), Contemporary Math. 237
(AMS, 1999) 4586
[abstract:] "We describe analogies between number theory zeta functions, dynamical zeta functions,and statistical mechanics zeta
functions, with emphasis on multivariable zeta functions. We mainly consider twovariable zeta functions $\zeta_{f}(z,s)$ in which
the variable $z$ is a "geometric variable", while the variable $s$ is an "arithmetic variable". The $s$variable has a thermodynamic
interpretation, in which $s$ parametrizes a family of energy functions $\phi_{s}$. We survey results on the analytic continuation and
location of zeros and poles of twovariable zeta functions for four examples connected with number theory. These examples are (1) the
beta transformation $f(x) = \beta x$ (mod 1), (2) the Gauss continued fraction map $f(x) = 1/x$ (mod 1), (3) zeta functions of varieties
over finite fields, and (4) Riemann zeta function."
V. Baladi, "A brief introduction to dynamical zeta functions", from
Classical Nonintegrability, Quantum Chaos (eds. A.
Knauf, Ya. Sinai) DMV Seminar, Vol 27. Basel (Birkhauser, 1997)
R. Mainieri, "Arithmetical properties of dynamical
zeta functions"
"The zeta functions we have been considering share more than a formal
resemblance to the Riemann zeta functions. It is possible to give an algebraic
structure to the set of periodic orbits, strengthening the analogy between
the thermodynamical zeta functions and the Riemann zeta functions...Upon
correct interpretation many of the results of multiplicative number theory
can be directly translated into equivalent statements about thermodynamical
zeta functions."
A. Voros, "Spectral zeta functions", in N. Kurokawa and T. Sunada,
editors, Zeta Functions in Geometry (Proceedings, Tokyo 1990),
Advanced Studies in Pure Mathematics 21 (1992) 327358.
W. Parry, "An analogue of
the prime number theorem for shifts of finite type and their suspensions",
Israel Journal of Mathematics 45 (1983) 4152.
[abstract:] "Following the classical procedure developed by Wiener and Ikehara
for the proof of the prime number theorem we find an asymptotic formula for the
number of closed orbits of a suspension of a shift of finite type when the
suspended flow is topologically weakmixing and when the suspending function is
locally constant."
This work was extended in Mark
Pollicott's Ph.D. thesis, supervised by Parry, and
led to the following paper:
W. Parry and M. Pollicott, "An analogue of the prime number theorem for
closed orbits of axiom A flows", Annals of Mathematics 118
(1983) 573591.
[abstract:] "For an axiom A flow restricted to a basic set we extend the zeta
function to an open set containing $\script{R}(s) \geq h$ where $h$ is the topological
entropy. This enables us to give an asymptotic formula for the number of closed orbits
by adapting the WienerIkehara proof of the prime number theorem."
This result was refined when the error term was considered (for certain cases) here:
M. Pollicott and R. Sharp, "Exponential error terms for growth functions on negatively curved surfaces", American Journal of Mathematics 120 (1998) 10191042.
[abstract:] "In this paper we consider two counting problems associated with compact negatively
curved surfaces and improve classical asymptotic estimates due to Margulis. In the first, we show that
the number of closed geodesics of length at most T has an exponential error term. In the second
we show that the number of geodesic arcs (between two fixed points x and y) of length
at most T has an exponential error term. The proof is based on a detailed study of the zeta
function and Poincaré series and benefits from recent work of Dolgopiat."
W. Parry and M. Pollicott, "Zeta functions and the periodic orbit
structure of hyperbolic dynamics", Asterisque (1990) 187188.
M. Pollicott, "Closed geodesics and zeta functions", from Ergodic Theory, Symbolic Dynamics and
Hyperbolic Spaces (eds.T. Bedford, et. al.) Oxford. (O.U.P., 1992)
M. Pollicott, "Periodic orbits and zeta functions",
Proc. Symp. Pure Math. 69 (2001) 409427.
M. Pollicott's
research page, including notes on "Geometry, Number Theory and Dynamical Systems"
Pollicott's CV
including a complete list of publications
V. Petkov, "Analytic
singularities of the dynamical zeta function", Nonlinearity 12 (1999) 16631681.
[Abstract:] "We study the dynamical zeta function Z_{D}(s) related to
the periodic trajectories of the billiard flow for several disjoint strictly convex bodies in
R^{3}. We show that the analytic properties of Z_{D}(s)
close to the line of absolute convergence Re[s] = s_{0} are similar to the
behaviour close to the line Re[s] = 1 of the inverse Q(s) = 1/R(s)
of the classical Riemann zeta function R(s)."
J. Hilgert and D. Mayer,
"The dynamical zeta function and transfer
operators for the KacBaker model"
"The KacBaker model describes a 1dim. classical lattice spin system with exponentially
fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to
investigate the phenomenon of phase transition in systems with weak longrange interactions like
van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of
Fredholm determinants of two transfer operators and hence is a meromorphic function. One of
the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the
Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator
acting in some Banach space of holomorphic observables of the system. In this paper we show
how the Kac operator can be explicitly related basically through the SegalBargmann transform
to the Ruelle operator restricted to a certain SegalBargmann space of entire functions in the
complex plane. This allows us to show that Ruelle's zeta function for the KacBaker model has
infinitely many "nontrivial" zeros on the real axis. In a special case we can show the existence of
also infinitely many "trivial" zeros on the line Re s = ln 2 in the complex splane. Hence some kind
of Riemann hypothesis seems to hold for this dynamical zeta function."
J. Hilgert and D.Mayer, "The dynamical zeta function and transfer
operators for a class of lattice spin models"
"We investigate the location of zeros and poles of a dynamical zeta function arising in a
class of lattice spin models introduced in the 60's by M. Kac. The transfer operator method
allows us to prove the xistence of infinitely nontrivial zeros of this function on the real
line. For certain parameter values there exist also infinitely many trivial equally spaced zeros
on a line parallel to the imaginary axis. Hence also for this kind of dynamical zeta function
some kind of Riemann hypothesis seems to hold."
J. Hilger, D. Mayer and H. Movasati, "Transfer
operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,Z)$"
(preprint, 03/03)
[abstract:] "In this article we report on a surprising relation between the transfer
operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators on the space
of period functions for the modular group $\PSL(2,Z)$. For this we study special
eigenfunctions of the transfer operators with eigenvalues +1, which are also
solutions of the Lewis equations for the groups $\Gamma_{0}(n)$ and which are determined by
eigenfunctions of the transfer operator for the modular group $\PSL(2,Z)$. In
the language of the AtkinLehner theory of old and new forms one should hence call them
old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the
components of these old solutions for the group $\Gamma_{0}(n)$ determine for any n a
solution of the Lewis equation for the modular group and hence also an eigenfunction of
the transfer operator for this group."
D. Mayer, "On a zeta function related to the continued fraction transformation",
Bull. Soc. Math. France 104 (1976) 195203.
M. Eisele and D.
Mayer, "Dynamical zeta functions for Artin's billiard and the VenkovZograf
factorization formula", Physica D 70, No. 4 (1994) 342356.
A. Momeni and A. Venkov, "Mayer transfer operator approach to Selberg zeta function" (preprint 08/2010)
[abstract:] "These notes are based on three lectures given by second author at Copenhagen University (October 2009) and at Aarhus University, Denmark (December 2009). We mostly present here a survey of results of Dieter Mayer on relations between Selberg and SmaleRuelle dynamical zeta functions. In a special situation the dynamical zeta function is defined for a geodesic flow on a hyperbolic plane quotient by an arithmetic cofinite discrete group. More precisely, the flow is defined for the corresponding unit tangent bundle. It turns out that the Selberg zeta function for this group can be expressed in terms of a Fredholm determinant of a classical transfer operator of the flow. The transfer operator is defined in a certain space of holomorphic functions and its matrix representation in a natural basis is given in terms of the Riemann zeta function."
A. Lopes, "The zeta function, nondifferentiability
of pressure, and the critical exponent of transition"
"The main purpose of this paper is to analyze the lack of differentiability
of the pressure and, from the behaviour of the pressure around the point
of nondifferentiability, to derive an asymptotic formula for the number
of periodic orbits of a dynamical system...This result follows from analysis
of the dynamic zeta function and Tauberian theorems. We introduce a functional
equation relating the pressure and the Riemann zeta function, and this
equation plays an essential role in the proof of our results."
A. Lopes and R. Markarian, "Open billiards: Invariant
and conditionally invariant probabilities on Cantor sets" (abstract)
"We analyze the dynamics of a class of billiards (the open billiard
on the plane) in terms of invariant and conditionally invariant probabilities.
The dynamical system has a horseshoe structure...A formula relating entropy,
Lyapunov exponent, and Hausdorff dimension of a natural probability
m
for the system is presented...As the system has a horseshoe structure,
one can compute the asymptotic growth rate of n(r), the number
of closed trajectories with the largest eigenvalue of the derivative smaller
than r. This theorem implies good properties for the poles of the
associated Zeta function and this result turns out to be very important
for the understanding of scattering quantum billiards."
A.L. Fel'shtyn,
"Dynamical zeta functions in Nielson theory and Reidemeister
torsion"
T. Ward, "Dynamical
zeta functions for typical extensions of full shifts"
"We consider a family of isometric extensions of the full shift on p
symbols (for p a prime) parametrized by a probability space. Using
HeathBrown's work on the Artin conjecture, it is shown that for all but
two primes p the set of limit points of the growth rate of periodic
points is infinite almost surely. This shows in particular that the dynamical
zeta fuction is not algebraic almost surely."
V. Chothi, G. Everest and T. Ward, "Sinteger dynamical
systems: periodic points" (abstract)
J. Bolte, C. Matthies, M. Sieber and F. Steiner, "Crossing
the entropy barrier of dynamical zeta functions", Physica D 63 (1993) 7186
"Dynamical zeta functions are an important tool to quantize chaotic
dynamical systems. The basic quantization rules require the
computation of the zeta functions on the real energy axis, where their
Euler product representations running over the classical periodic
orbits usually do not converge due to the
existence of the socalled entropy barrier determined by the
topological entropy of the classical system. We show that the
convergence properties of the dynamical zeta functions rewritten as
Dirichlet series are governed not only by the wellknown topological
and metric entropy, but depend crucially on subtle statistical
properties of the Maslov indices and of the multiplicities of
the periodic orbits that are measured by a new parameter for which
we introduce the notion of a third entropy. If and only if
the third entropy is nonvanishing, one can cross the entropy
barrier; if it exceeds a certain value, one can even
compute the zeta function in the physical region by means of a
convergent Dirichlet series. A simple statistical model is
presented which allows to compute the third entropy.
Four examples of chaotic systems are studied in detail to test the
model numerically."
A. Juhl,
"Secondary invariants and the singularity of the Ruelle zetafunction in the
central critical point"
W.T. Lu and S. Sridhar, "Correlations among the Riemann zeros:
Invariance, resurgence, prophecy and selfduality" (preprint 05/04)
[abstract:] "We present a conjecture describing new long range correlations
among the Riemann zeros leading to 3 principal features: (i) The spectral
autocorrelation is invariant with respect to the averaging window. (ii) Resurgence
occurs wherein the lowest zeros appear in all autocorrelations. (iii) Suitably
defined correlations lead to predictions (prophecy) of new zeros. This conjecture
is supported by analytical arguments and confirmed by numerical calculations using
10^{22} zeros computed by Odlyzko. The results lead to a selfduality of
the Riemann spectrum similar to the quantumclassical duality observed in billiards."
See section V, where Ruelle zeta function are discussed:
"The motivation for this work has come from experimental observations
in the microwave transmission of open ndisk billiards which led
to the observation of classical RuellePollicott resonances in the autocorrelation
of quantum spectra of hyperbolic ndisk open billiards. The result
established a new approach to quantumclassical correspondence by
demonstrating a correspondece between the quantum and classical resonance
spectra of an open chaotic system. Applying the same procedures developed
there to the Riemann spectrum, we have arrived at the results described
in this paper..."
P. Cvitanovic and H. Rugh, "A Fredholm determinant for semiclassical quantization"
[abstract:] "We investigate a new type of approximation to quantum determinants, the "quantum
Fredholm determinant", and test numerically the conjecture that for Axiom A hyperbolic
flows such determinants have a larger domain of analyticity and better convergence than the
GutzwillerVoros zeta functions derived from the Gutzwiller trace formula. The conjecture
is supported by numerical investigations of the 3disk repeller, a normalform model of a
flow, and a model 2d map."
R.E. Crandall, "On the quantum zeta function", Journal of Physics A 29
(1996) 67956816.
[abstract:] "It is remarkable that the quantum zeta function, defined as a sum over energy
eigenvalues E:
$Z(s) = \sum \frac{1}{E^s}$
admits of exact evaluation in some situations for which not a single E be known.
Herein we show how to evaluate instances of Z(s), and of an associated parity
zeta function Y(s), for various quantum systems. For some systems both Z(n),
Y(n) can be evaluated for infinitely many integers n. Such Z,
Y values can be used, for example, to effect sharp numerical estimates of a system's ground energy.
The difficult problem of evaluating the analytic continuation Z(s) for arbitrary
complex s is discussed within the contexts of perturbation expansions, path integration,
and quantum chaos.
M.V. Berry, "Spectral zeta functions for
AharanovBohm quantum billiards", Journal of Physics A
19 (1986) 22812296.
M.L. Lapidus and M. van Frankenhuysen, "A prime orbit theorem for
selfsimilar flows and Diophantine approximation", Contemporary
Mathematics volume 290 (AMS 2001) 113138.
"Assuming some regularity of the dynamical zeta function, we
establish an explicit formula with an error term for the prime orbit
counting function of a suspended flow. We define the subclass of
selfsimilar flows, for which we give an extensive analysis of the error
term in the corresponding prime orbit theorem...The precise order of
the error term depends on the 'dimension free' region of the dynamical
zeta function, as in the classical Prime Number Theorem. This region
in turn depends on properties of Diophantine approximation of the
weights of the flow."
A. Teplyaev, "Spectral zeta functions of
fractals and the complex dynamics of polynomials" (preprint 05/05)
[abstract:] "We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals,
such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function
associated with a polynomial. It is proved that this zeta function has a meromorphic continuation to a half plain with poles
contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta functions of a
quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket
is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of
the latter. A similar product structure was discovered by M.L. Lapidus for selfsimilar fractal strings."
S. Egger né Endres and F. Steiner, "An exact trace formula and zeta functions for an infinite quantum graph with a nonstandard Weyl asymptotics" (preprint 04/2011)
[abstract:] "We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a oneparameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show that the oneparameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet Laplacian is the limit of the oneparameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the nonstandard Weyl asymptotics $\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$ for the eigenvalue counting function. Based on an exact trace formula, the Vorono\"i summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a welldefined (renormalized) secular equation and a Selberglike zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true."
O. Bohigas, P. Leboeuf, and
M.J. Sanchez,
"Spectral spacing correlations for chaotic and disordered systems"
"New aspects of spectral fluctuations of (quantum) chaotic and
diffusive systems are considered, namely autocorrelations of the
spacing between consecutive levels or spacing autocovariances. They
can be viewed as a discretized two point correlation function. Their
behavior results from two different contributions. One corresponds
to (universal) random matrix eigenvalue fluctuations, the other to
diffusive or chaotic characteristics of the corresponding classical
motion. A closed formula expressing spacing autocovariances in terms
of classical dynamical zeta functions, including the PerronFrobenius
operator, is derived. It leads to a simple interpretation in terms of
classical resonances. The theory is applied to zeros of the Riemann
zeta function. A striking correspondence between the associated
classical dynamical zeta functions and the Riemann zeta itself is
found. This induces a resurgence phenomenon where the lowest Riemann
zeros appear replicated an infinite number of times as resonances and
subresonances in the spacing autocovariances. The theoretical results
are confirmed by existing "data". The present work further extends
the already well known semiclassical interpretation of properties of
Riemann zeros."
G. Lambiase, V.V. Nesterenko, M. Bordag,
"Casimir energy of a ball and cylinder in the zeta function technique"
"A simple method is proposed to construct the spectral zeta functions
required for calculating the electromagnetic vacuum energy with boundary
conditions given on a sphere or on an infinite cylinder...The starting
point of the consideration is the representation of the zeta functions
in terms of contour integral, further the uniform asymptotic expansion
of the Bessel function is essentially used. After the analytic continuation,
needed for calculating the Casimir energy, the zeta functions are presented
as infinite series containing the Riemann zeta function with rapidly falling
down terms."
V.Nesterenko and I. Pirozhenko,
"Spectral zeta functions for a cylinder and a circle"
O. Agam, A.V. Andreev, B.L. Altshuler,
"Relations between quantum and classical spectral determinants
(zetafunctions)"
"We demonstrate that beyond the universal regime correlators of quantum
spectral determinants $\Delta(\epsilon)=\det (\epsilon\hat{H})$ of
chaotic systems, defined through an averaging over a wide energy interval,
are determined by the underlying classical dynamics through the spectral
determinant $1/Z(z)=\det (z {\cal L})$, where $e^{{\cal L}t}$ is
the PerronFrobenius operator. Application of these results to the Riemann
zeta function, allows us to conjecture new relations satisfied by this
function."
A class of zeta functions that extends the class of Epstein's has been
brought to my attention by Emilio Elizalde. They
are spectral zeta functions associated with a quadratic + linear + constant
form in any number of dimensions. Elizalde has developed formulas
for them which extend the famous ChowlaSelberg formula.
E. Elizalde, "Spectral
zeta functions in noncommutative spacetime"
E. Elizalde, "Explicit zeta functions for bosonic and fermionic
fields on a noncommutative toroidal spacetime", Journal of Physics
A 34 (2001) 30253036.
E. Elizalde, "Multidimensional extension of the generalized
ChowlaSelberg formula", Communications in Mathematical Physics
198 91998) 8395.
E. Elizalde, "Zeta functions, formulas and applications",
J. Comp. Appl. Math. 118 (2000) 125.
Chapter 1 of Elizalde's Ten Physical Applications
of Spectral Zeta Functions (1995)
"In this introductory chapter, an overview of the method of zeta function regularization
is presented. We start with some brief historical considerations and by introducting the
specific zeta functions that will be used in the following chapters in a number of physical
situations. We summarize the basic properties of the different zeta functions. We define
the concept of zeta function associated with an elliptic partial differential operator, and
point towards its uses to define 'the determinant' of the operator. We show explicitly how
to regularize the Casimir energy in some simple cases in a correct way, thereby introducing
the zetafunction
regularization procedure. Finally, these fundamental concepts are both extended and made
much more precise in the last section, where examples of the most recent developments on
poweful applications of the theory are discussed."
M.L. Lapidus, "Spectral and fractal geometry: From the WeylBerry conjecture
for the vibrations of fractal drums to the Riemann zetafunction" (from
Ordinary and Partial Differential Equations and Mathematical Physics,
Ed. C. Bennewitz  Proceedings of the Fourth UAB International Conference,
Birmingham 1990) (Academic Press, 1992) 151182.
M.L. Lapidus, "Vibrations of fractal drums, the Riemann hypothesis, waves
in fractal media, and the WeylBerry conjecture" (from Ordinary and
Partial Differential Equations, Eds. B. Sleeman, et. al.  volume
IV, Proceedings of the Twelfth International Conference, Dundee 1992) (Pitman
Research Notes in Mathematics Series 289, Longman Scientific and
Technical, 1993) 126209.
M.L. Lapidus and C. Pomerance, "The Riemann zetafunction and the
onedimensional WeylBerry conjecture for fractal drums", Proceedings of
the London Mathematical Society (3) 66 (1993) 4169.
"Based on his earlier work on the vibrations of 'drums with fractal
boundary', the first author has refined M.V. Berry's conjecture that
extended from the 'smooth' to the 'fractal' case H. Weyl's conjecture
for the asymptotics of the eigenvalues of the Laplacian on a bounded
open subset of R^{n}. We solve here in the
onedimensional case (that is n = 1) this 'modified WeylBerry
conjecture'. We discover, in the process, some unexpected and
intriguing connections between spectral geometry, fractal geometry
and the Riemann zetafunction. we therefore show that one can 'hear'
(that is, recover from the spectrum) not only Minkowski fractal
dimension of the boundary  as was established previously by the
first author  but also, under the stronger assumptions of the
conjecture, its Minkowski content (a 'fractal' analogue of its
'length').
We also prove (still in dimension one) a related conjecture of the
first author, as well as its converse, which characterizes the
situation when the error estimates of the aforementioned paper are
sharp."
M.L. Lapidus and C. Pomerance, "Counterexamples to the modified WeylBerry
conjecture on fractal drums", Mathematical Proceedings of the Cambridge
Philosophical Society 119 (1996) 167178.
M.L. Lapidus and C. Pomerance, "Fonction zeta de Riemann et conjecture de
WeylBerry pour les tambours fractals", C. R. Acad. Sci. Paris Ser. I
Math. 310 (1990) 343348.
M.L. Lapidus and H. Maier, "Hypothese de Riemann, cordes fractales vibrantes
et conjecture de WeylBerry modifiee", C. R. Acad. Sci Paris Ser. I Math.
313 (1991) 1924.
M.L. Lapidus and H. Maier, "The Riemann Hypothesis and inverse spectral
problems for fractal strings", Journal of the London Mathematical Society
(second edition) 52 (1995) 1534.
"Motivated in part by the first author's work on the WeylBerry
conjecture for the vibrations of 'fractal drums' (that is, 'drums with
fractal boundary'), M.L. Lapidus and C. Pomerance have studied a
direct spectral problem for the vibrations of 'fractal strings' (that
is, onedimensional 'fractal drums') and establihed in the process
some unexpected connections with the Riemann zetafunction in the
'critical interval' 0 < s < 1. In this paper we show, in
particular, that the converse of their theorem (suitably interpreted as a
natural inverse spectral problem for fractal strings, with boundary of
Minkowski fractal dimension D in (0,1)) is not true in the
'midfractal' case when D = 1/2, but that it is true for all
other D in the critical interval (0,1) if and only if
the Riemann hypothesis is true. We thus obtain a new characterization
of the Riemann hypothesis by means of an inverse spectral problem.
(Actually, we prove the following stronger result: for a given
D in (0,1), the above inverse spectral problem is equivalent
to the 'partial Riemann hypothesis' for D, according to which
[the Riemann zeta function] does not have any zero
on the vertical line Re s = D.) Therefore, in some very
precise sense, our work shows that the question (a la Marc Kac)
"Can one hear the shape of a fractal string?"  now interpreted as a
suitable converse (namely, the above inverse problem)  is intimately
connected with the existence of zeros of [the Riemann zeta function]
in the critical strip 0 < Re s < 1, and
hence to the Riemann hypothesis."
M.L. Lapidus and M. van Frankenhuysen, "Complex dimensions of fractal strings
and explicit formulas for geometric and spectral zetafunctions", Preprint
IHES/M/97/34, Institut des Hautes Etudes Scientifiques, BuressurYvette,
France, April 1997.
M.L. Lapidus and M. van Frankenhusen, "Complex dimensions of fractal
strings and oscillatory phenomena in fractal geometry and arithmetic",
from Spectral Problems in Geometry and Arithmetic (T. Branson,ed.),
Contemporary Mathematics, vol. 237 (AMS, 1999) 87105.
"We put the theory of Dirichlet series and integrals in the
geometric setting of 'fractal strings' (onedimensional drums with
fractal boundary). The poles of a Dirichlet series thus acquire the
geometric meaning of 'complex dimensions' of the associated fractal
string, and they describe the geometric and spectral oscillations of
this string by means of an 'explicit formula'. We define the
'spectral operator', which allows us to characterize the presence of
critical zeros of zetafunctions from a large class of Dirichlet
series as the questions of invertibility of this operator. We thus
obtain a geometric reformulation of the generalized Riemann
Hypothesis, thereby extending the earlier work of the first author
with H. Maier. By considering the restriction of this operator to the
subclass of 'generalized Cantor strings', we prove that zetafunctions
from a large subclass of this class have no infinite sequence of zeros
forming a vertical arithmetic progression. (For the special case of
the Riemann zetafunction, this is Putnam's theorem.)
We make an extensive study of the complex dimensions of
'selfsimilar' fractal strings, to gain further insight into the kind
of geometric infromation contained in the complex dimensions. We also
obtain a formula for the volume of the tubular neighborhoods of a
fractal string and draw an analogy with Riemannian geometry. Our
work suggests to define 'fractality' as the presence of nonreal
complex dimensions with positive real part."
N. Lal and Michel L. Lapidus, "Higherdimensional complex dynamics and spectral zeta functions of fractal differential Sturm–Liouville operators" (preprint 02/2012)
[abstract:] "We investigate the spectral zeta function of a selfsimilar Sturm–Liouville operator associated with a fractal selfsimilar measure on the halfline and C. Sabot's work connecting the spectrum of this operator with the iteration of a rational map of several complex variables. We obtain a factorization of the spectral zeta function expressed in terms of the zeta function associated with the dynamics of the corresponding renormalization map, viewed as a rational function on the complex projective plane. The result generalizes to several complex variables and to the case of fractal Sturm–Liouville operators a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended to the Sierpinski gasket and some other decimable fractals by A. Teplyaev. As a corollary, in the very special case when the underlying selfsimilar measure is Lebesgue measure on $[0,1]$, we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev. The above fractal Hamiltonians and their spectra are relevant to the study of diffusions on fractals and to aspects of condensed matters physics, including to the key notion of density of states."
books
M.L. Lapidus and M. van Frankenhuysen, eds., Dynamical,
Spectral, and Arithmetic ZetaFunctions, Contemporary Mathematics volume 290 (AMS, 2001)
"This volume grew out of the special session on dynamical, spectral, and arithmetic zeta
functions held at the annual meeting of the American Mathematical Society in San Antonio,
but also includes four articles that were invited to be part of the collection. The purpose
of the meeting was to bring together leading researchers, to find links and analogies
between their fields, and to explore new methods. The papers discuss dynamical systems,
spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as
well as computational work on the Riemann zeta function."
A. Juhl,
Cohomological Theory of Dynamical Zeta Functions (Progress in Mathematics,
Vol. 194.) (Birkhauser, 2001)
D. Ruelle,
Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval
(CRM Monograph, Vol 4) (AMS, 1994)
"A monograph based on the Aisenstadt lectures given by the author in October 1993 at the
University of Montreal on "Dynamical Zeta Functions," but with a different emphasis.
Hyperbolic systems are not discussed in detail. After a general introduction (chapter 1),
the concentration is on piecewise monotone maps of the interval, and a detailed proof is
given of a generalized form of the theorem of Baladi and Keller (chapter 2)."
E. Elizalde,
Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in
Physics. New Series M, Monographs, M35 (SpringerVerlag, 1995) [Chapter 1]
"Zetafunction regularization is a powerful method in perturbation theory.
This book is meant as a guide for the student of this subject. Everything is
explained in detail, in particular the mathematical difficulties and tricky
points, and several applications are given to show how the procedure works in
practice (e.g. Casimir
effect, gravity and string theory, hightemperature
phase transition, topological symmetry breaking). The formulas some of which
are new can be used for accurate numerical calculations. The book is to be
considered as a basic introduction and a collection of exercises for those
who want to apply this regularization procedure in practice."
E. Elizalde, S.D. Odintsov, A. Romeo, S. Zerbini,
Zeta Regularization Techniques With Applications (World Scientific,
1994)
M.B. Schiekel, Zetafunktionen in der Physik (monograph, University of Ulm, 2011)
[abstract:] "Zeta functions in Physics  an Introduction" is an introductory monograph about Riemann zeta function, spectral zeta functions and dynamical zeta functions in physics on an advanced undergraduate level. Topics are: harmonic oscillator (not much surprising), phase operators and coherent states, path integrals of single and manyparticle systems,
regularization with spectral zeta function and Casimir effect, heat kernel expansion and regularization and renormalization of the Phi(4,4)theory in 1loop approximation, dynamical or Ruelle zeta functions in dynamical systems and Gutzwiller's trace formula. Mathematical appendices deal with: Gamma function, Riemann zeta function, Mellin transformation, asymptotic expansions, heat kernel expansion and (analytic) index theorem for elliptic differential operators, Fredholm theory of elliptic pseudodifferential operators. Due to the
didactic intention of this introduction all proofs are worked out in detail. (Text in German)"
Thermodynamic formalism and dynamic zeta functions, DFG research group, TU Clausthal, Germany.
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