## 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 (Yang-Baxter, 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) 82-99.

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, "L-series of a covering", Proceedings of the National Academy of Sciences 42 No. 7 (1956) 422-424

From http://www.math-phys-zeta.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) 212-214.

[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."

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) 45-86

[abstract:] "We describe analogies between number theory zeta functions, dynamical zeta functions,and statistical mechanics zeta functions, with emphasis on multi-variable zeta functions. We mainly consider two-variable 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 two-variable 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)

"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) 327-358.

W. Parry, "An analogue of the prime number theorem for shifts of finite type and their suspensions", Israel Journal of Mathematics 45 (1983) 41-52.

[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 weak-mixing 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) 573-591.

[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 Wiener-Ikehara 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) 1019-1042.

[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) 187-188.

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) 409-427.

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) 1663-1681.

[Abstract:] "We study the dynamical zeta function ZD(s) related to the periodic trajectories of the billiard flow for several disjoint strictly convex bodies in R3. We show that the analytic properties of ZD(s) close to the line of absolute convergence Re[s] = s0 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)."

"The Kac-Baker model describes a 1-dim. 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 long-range 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 Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many "non-trivial" 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 s-plane. 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 Atkin-Lehner 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) 195-203.

M. Eisele and D. Mayer, "Dynamical zeta functions for Artin's billiard and the Venkov-Zograf factorization formula", Physica D 70, No. 4 (1994) 342-356.

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 Smale-Ruelle 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."

"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 non-differentiability, 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."

"We consider a family of isometric extensions of the full shift on p symbols (for p a prime) parametrized by a probability space. Using Heath-Brown'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, "S-integer 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) 71-86

"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 so--called 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 well--known 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."

W.T. Lu and S. Sridhar, "Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality" (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 auto-correlation is invariant with respect to the averaging window. (ii) Resurgence occurs wherein the lowest zeros appear in all auto-correlations. (iii) Suitably defined correlations lead to predictions (prophecy) of new zeros. This conjecture is supported by analytical arguments and confirmed by numerical calculations using 1022 zeros computed by Odlyzko. The results lead to a self-duality of the Riemann spectrum similar to the quantum-classical 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 n-disk billiards which led to the observation of classical Ruelle-Pollicott resonances in the auto-correlation of quantum spectra of hyperbolic n-disk open billiards. The result established a new approach to quantum-classical 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 semi-classical 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 Gutzwiller-Voros zeta functions derived from the Gutzwiller trace formula. The conjecture is supported by numerical investigations of the 3-disk repeller, a normal-form model of a flow, and a model 2-d map."

R.E. Crandall, "On the quantum zeta function", Journal of Physics A 29 (1996) 6795-6816.

[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 Aharanov-Bohm quantum billiards", Journal of Physics A 19 (1986) 2281-2296.

M.L. Lapidus and M. van Frankenhuysen, "A prime orbit theorem for self-similar flows and Diophantine approximation", Contemporary Mathematics volume 290 (AMS 2001) 113-138.

"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 self-similar 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 self-similar fractal strings."

S. Egger né Endres and F. Steiner, "An exact trace formula and zeta functions for an infinite quantum graph with a non-standard 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 one-parameter 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 one-parameter 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 one-parameter 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 non-standard 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 well-defined (renormalized) secular equation and a Selberg-like 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 Perron-Frobenius 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 sub-resonances 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 (zeta-functions)"

"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 Perron-Frobenius 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 Chowla-Selberg formula.

E. Elizalde, "Explicit zeta functions for bosonic and fermionic fields on a noncommutative toroidal spacetime", Journal of Physics A 34 (2001) 3025-3036.

E. Elizalde, "Multidimensional extension of the generalized Chowla-Selberg formula", Communications in Mathematical Physics 198 91998) 83-95.

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 zeta-function 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 Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function" (from Ordinary and Partial Differential Equations and Mathematical Physics, Ed. C. Bennewitz - Proceedings of the Fourth UAB International Conference, Birmingham 1990) (Academic Press, 1992) 151-182.

M.L. Lapidus, "Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry 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) 126-209.

M.L. Lapidus and C. Pomerance, "The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums", Proceedings of the London Mathematical Society (3) 66 (1993) 41-69.

"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 Rn. We solve here in the one-dimensional case (that is n = 1) this 'modified Weyl-Berry conjecture'. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. 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 Weyl-Berry conjecture on fractal drums", Mathematical Proceedings of the Cambridge Philosophical Society 119 (1996) 167-178.

M.L. Lapidus and C. Pomerance, "Fonction zeta de Riemann et conjecture de Weyl-Berry pour les tambours fractals", C. R. Acad. Sci. Paris Ser. I Math. 310 (1990) 343-348.

M.L. Lapidus and H. Maier, "Hypothese de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiee", C. R. Acad. Sci Paris Ser. I Math. 313 (1991) 19-24.

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) 15-34.

"Motivated in part by the first author's work on the Weyl-Berry 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, one-dimensional 'fractal drums') and establihed in the process some unexpected connections with the Riemann zeta-function 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 zeta-functions", Preprint IHES/M/97/34, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 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) 87-105.

"We put the theory of Dirichlet series and integrals in the geometric setting of 'fractal strings' (one-dimensional 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 zeta-functions 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 zeta-functions 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 zeta-function, this is Putnam's theorem.)

We make an extensive study of the complex dimensions of 'self-similar' 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, "Higher-dimensional complex dynamics and spectral zeta functions of fractal differential Sturm–Liouville operators" (preprint 02/2012)

[abstract:] "We investigate the spectral zeta function of a self-similar Sturm–Liouville operator associated with a fractal self-similar measure on the half-line 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 self-similar 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 Zeta-Functions, 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."

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 (Springer-Verlag, 1995)   [Chapter 1]

"Zeta-function 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, high-temperature 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 many-particle systems, regularization with spectral zeta function and Casimir effect, heat kernel expansion and regularization and renormalization of the Phi(4,4)-theory in 1-loop 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 pseudo-differential 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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