Showing papers on "Riemann zeta function published in 1991"

Periodic orbit expansions for classical smooth flows

Abstract: The authors derive a generalized Selberg-type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow. This relation is a classical analogue of the quantum trace formulae and Selberg-type zeta functions.

zeta -function regularization of one-loop effective potentials in anti-de Sitter spacetime.

Abstract: We use the $\ensuremath{\zeta}$-function technique to calculate the one-loop effective potential for a scalar field in anti-de Sitter (AdS) space. The $\ensuremath{\zeta}$ function is computed exactly on the four-dimensional hyperbolic space ${H}^{4}$, the Euclidean section appropriate for AdS space. The structure of the ultraviolet divergences is shown to agree with previous calculations where Pauli-Villars or some version of dimensional regularization was used. The finite part of the effective potential is given explicitly by an integral over a variable related to the spectrum of the Laplace-Beltrami operator on ${H}^{4}$.

On lognormal random variables: I-the characteristic function

Abstract: The characteristic function of a lognormal random variable is calculated in closed form as a rapidly convergent series of Hermite functions in a logarithmic variable. The series coefficients are Nielsen numbers, defined recursively in terms of Riemann zeta functions. Divergence problems are avoided by deriving a functional differential equation, solving the equation by a de Bruijn integral transform, expanding the resulting reciprocal Gamma function kernel in a series, and then invoking a convergent termwise integration. Applications of the results and methods to the distribution of a sum of independent, not necessarily identical lognormal variables are discussed. The result is that a sum of lognormals is distributed as a sum of products of lognormal distributions. The case of two lognormal variables is outlined in some detail.

Quantization of chaos.

Abstract: We study a rule for quantizing chaos based on the dynamical zeta function defined by a Euler product over the classical periodic orbits as suggested by Gutzwiller's semiclassical trace formula. A test of our approximate quantization formula is carried out for the planar hyperbold billiard, which shows that at least the first 150 quantum energy levels can be generated.

Asymptotics of a τ-function arising in the two-dimensional Ising model

Abstract: The short-distance assymptotics of the τ-function associated to the 2-point function of the two-dimensional Ising model is computed as a function of the integration constant defined from the long-distance behavior of the τ-function. The result is expressible in terms of the Barnes double gamma function (equivalently, the BarnesG-function).

Inhomogeneous multidimensional Epstein zeta functions

Abstract: The pole structure of the inhomogeneous multidimensional Epstein zeta function, Em2N(s; a1,...,aN)=∑∞n1,...,nN =1 (a1n21+⋅⋅⋅+aNn2N +m2)−s, is determined using heat‐kernel techniques. The poles of Em2N(s; a1,...,aN) are found to be s=N/2; (N−1)/2;...; (1)/(2) ; −(2l+1)/2, l∈ N0. Furthermore, their residues and Em2N(−p; a1,...,aN), p∈ N0, are given explicitly. These results are used to find the high‐temperature expansion of the Helmholtz free‐energy of a massive spin‐0 and spin‐ (1)/(2) gas subject to Dirichlet boundary conditions on hypercuboids in a flat n‐dimensional space‐time.

Casimir effect at finite temperature

Abstract: The high-temperature expansion of the grand thermodynamic potential of a nonconformally invariant spin-0 gas in an arbitrary ultrastatic spacetime with boundary is given in terms of the Minakshisundaram-Pleijel coefficients of the heat-kernel and the zeta function of the spatial section. The general formula is then used to find the expansion in the case of a massive bosonic field subject to Dirichlet boundary conditions on hypercuboids in a flat n-dimensional spacetime. A detailed analysis of inhomogeneous multidimensional Epstein zeta functions is necessary and some new properties of them are derived. Finally the thermodynamics of the system is considered.

Generalized Riemann zeta -function regularization and Casimir energy for a piecewise uniform string.

Abstract: The generalized $\ensuremath{\zeta}$-function techniques will be utilized to investigate the Casimir energy for the transverse oscillations of a piecewise uniform closed string. We find that the $\ensuremath{\zeta}$-function regularization method can lead straightforwardly to a correct result.

Determinant expression of Selberg zeta functions. II

Abstract: We will prove that for PSL(2, R) and its cofinite subgroup, the Selberg zeta function is expressed by the determinant of the Laplacian. We will also give an explicit calculation in case of congruence subgroups, and deduce that the part of the determinant of the Laplacian composed of the continuous spectrum is expressed by Dirichlet ?-functions. The first discovery of the relation between the Selberg zeta function and the determinant of the Laplacian was by physicists (3, 4, 7). Sarnak (14) and Voros ( 15) obtained the determinant expression of Selberg zeta functions for compact Riemann surfaces with torsionfree fundamental groups. In those cases, all the spectrum of the Laplacians are discrete. The determinant was defined via the holomorphy at the origin of the spectral zeta function of Minakshisundaram and Pleijel (13). For noncompact but finite Riemann surfaces with torsion- free fundamental groups, these results are generalized by Efrat (5). In this case there exist both discrete and continuous spectrum. He constructs the spectral zeta function composed not only of eigenvalues but some values concerning continuous spectrum, which are decided by all the poles of the scattering de- terminant in the Selberg trace formula. The determinant of the Laplacian is defined by the standard method with the holomorphy of the spectral zeta func- tion at the origin. The aim of the present paper is to generalize his results to the case with any fundamental group Y (§3) and to give some arithmetic examples of the determinant of the Laplacians (§4). In §4, we restrict ourselves to the case when Y is a congruence subgroup of PSL(2, Z). In this case the partial spectral zeta function composed of only eigenvalues is also holomorphic at the origin (11, Theorem 3.3). Hence we have a decomposition of the determinant into parts corresponding to the discrete and continuous spectrum. The scatter- ing determinant is expressed very explicitly by Huxley (8) in terms of Dirichlet 7-functions £(i, x) ■ Almost all the poles of the scattering determinant are de- scribed by nontrivial zeros of L(s, x) ■ The continuous part of the determinant

Generalized local and global Sebastiani-Thom type theorems

Abstract: © Foundation Compositio Mathematica, 1991, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Curiosities of arithmetic gases

Abstract: Statistical mechanical systems with an exponential density of states are considered. The arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson‐parafermion equivalence is obtained using properties of the Riemann zeta function. Interactions (nontrivial mixing) among arithmetic gases using the concept of twisted convolutions are also introduced. Examples of exactly solvable models are discussed in detail.

Determinant of the Neumann operator on smooth Jordan curves

Abstract: Using the method of heat kernel expansion, the determinant of the Neumann operator on an arbitrary smooth Jordan curve is shown to be equal to the circumference.

Periodic orbit quantization of bound chaotic systems

Abstract: The authors present periodic orbit quantizations of the hyperbola billiard and the x2y2 potential. These two systems may be considered as belonging to the one-parameter family of potentials (x2y2)1a/. The quantum states are determined by means of the zeros of an expanded and truncated Selberg zeta function. The symmetries of the problem are considered and the Selberg zeta function is factorized into the irreducible representations of the symmetry group. The thus calculated eigenenergies are in good agreement with quantum mechanical calculations and converge when the number of terms in the expansion is increased. The results strongly indicate that the trace formula provides individual quantum eigenstates for chaotic systems.

Logarithmic decay and overconvergence of the unit root and associated zeta functions

Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1991, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Epstein zeta function and toy p-branes from old string action

Abstract: The authors study toy p-branes from old string action, and regularize their Casimir energies by the Epstein zeta function. They obtain the special dimensions for the open (p=2, 3, 4) and closed (p=2, 3, 4, 6, 8) branes with equal sides. The upper bound of p is obtained. They also discuss the special dimensions for open and closed membranes with unequal sides. The resulting dimensions may be integers for the case of unequal sides.