Epstein Zeta Function Research Articles

A restricted Epstein zeta function and the evaluation of some definite integrals

A restricted Epstein zeta function and the evaluation of some definite integrals

Rigorous proof of the attractive nature for the Casimir force of ap-odd hypercube

The Casimir effect giving rise to an attractive force between the configuration boundaries that confine the massless scalar field is rigorously proven for odd dimensional hypercube with the Dirichlet boundary conditions and different spacetime dimensions D by the Epstein zeta function regularization.

Values of the Epstein zeta functions at the critical points

Values of the Epstein zeta functions at the critical points

The height of the Leech lattice

We derive explicit formulae for the height of the Leech lattice and give evidence supporting the conjecture that the Leech lattice has minimum height and minimum values of the Epstein zeta function among all 24 dimensional lattices of determinant one.

Zeta functions: formulas and applications

The existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed, as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the Wodzicki residue. Exponentially fast convergent formulas – valid in the whole of the complex plane and yielding the pole positions and residua – that extend the ones by Chowla and Selberg for the Epstein zeta function (quadratic form) and by Barnes (affine form) are then given. After briefly recalling the zeta function regularization procedure in quantum field theory, some applications of these expressions in physics are described.

Phase structure of Gross-Neve model, taking the influence of temperature and finite volume into account

A regular procedure (based on the Riemann — Epstein zeta function) is proposed for calculating the effective potential of the Gross — Neve model in the case of a two-dimensional lattice, with various boundary conditions modeling the effects of finite volume and temperature. The effective potential and phase structures of the model are constructed for various boundary conditions. In the two-dimensional case, in contrast to the three-dimensional case, the phase pattern does not depend on the coupling constants.

A minimum for the theta function in three variables and the solution of the Rankin-Sobolev problem in a three-dimensional space

In this paper, an absolute minimum is found for the specialized theta function defined by a positive-definite ternery quadratic form with real coefficients. The result obtained yields an absolute minimum for the Epstein zeta function of the corresponding form. Bibliography: 15 titles.

On the vacuum stability in the Efimov–Fradkin model at finite temperature

The behavior of the nontruncated and truncated Efimov–Fradkin models (ℒint=−∑Nn=3λnφn) at finite temperature in a generic D-dimensional flat space–time was investigated. The thermal contribution to the renormalized mass and coupling constants are obtained in the one-loop approximation by the use of a mix between dimensional and the Epstein zeta function analytic regularization and a modified minimal subtraction procedure. We proved that for Dc(N−1)≤D there is not a temperature for which at least one of the renormalized coupling constants becomes zero, where Dc(N−1) is the critical spacetime dimension for the renormalized coupling constant λN−1. For Dc(N)≤D<Dc(N−1) only the renormalized coupling constant λN−1 becomes zero at some temperature β−1N−1. For D<Dc(N) the renormalized coupling constants λN−1(β) and λN(β) become zero at temperatures β−1N−1 and β−1N, respectively. In the latter situation, for temperatures β−1N−1<β−1<β−1N the effective potential has a global minimum. For temperatures above β−1N, the system can develop a first order phase transition, where the origin corresponds to a metastable vacuum. In the nontruncated model, corresponding to a nonpolynomial Lagrange density, for D≥2 all the coupling constants remain positive for any temperature.

Spectral representation of the spacetime structure: The "distance" between universes with different topologies.

We investigate the representation of the geometrical information of the Universe in terms of spectra, i.e., a set of eigenvalues of the Laplacian defined on the Universe. Here, we concentrate only on one specific problem along this line: to introduce a concept of distance between universes in terms of the difference in the spectra. We can find such a measure of closeness from a general discussion. First, we introduce a suitable functional ${\mathit{P}}_{\mathcal{G}}$[\ensuremath{\cdot}], where the geometrical information G (represented by the spectra) determines the detailed shape of the functional. Then, the overlapping functional integral between ${\mathit{P}}_{\mathcal{G}}$[\ensuremath{\cdot}] and ${\mathit{P}}_{\mathcal{G}\ensuremath{'}}$[\ensuremath{\cdot}] is taken, providing a measure of closeness between G and G\ensuremath{'}, d(G,G\ensuremath{'}). The basic properties of this distance (hereafter referred to as ``spectral distance,'' for brevity) are then investigated. First, it can be related to a reduced density-matrix element in quantum cosmology between G and G\ensuremath{'}. Thus, calculating the spectral distance d(G,G\ensuremath{'}) gives us insight into the quantum theoretical decoherence between two universes, corresponding to G and G\ensuremath{'}. Second, the spectral distance becomes divergent except for when G and G\ensuremath{'} have the same dimension and volume. This is very suggestive if the above-mentioned density-matrix interpretation is taken into account. Third, d(G,G\ensuremath{'}) does not satisfy the triangular inequality, which illustrates clearly that the spectral distance and the distance defined by the DeWitt metric on the superspace are not equivalent. We then pose a question: Do two universes with different topologies interfere with each other quantum mechanically? In particular, we concentrate on the difference in the orientabilities. To investigate this problem, several concrete models in two dimensions are set up, and the spectral distances between them are investigated: distances between tori and Klein's bottles, and those between spheres and real projective spaces. Quite surprisingly, we find many cases of spaces with different orientabilities in which the spectral distance turns out to be very short. This may suggest that, without any other special mechanism, two such universes interfere with each other quite strongly, contrary to our intuition. We discuss some curious features of the heat kernel for tori and Klein's bottles in terms of Epstein's theta and zeta functions. Differences and parallelisms between the spectral distance and the DeWitt distance are also discussed. \textcopyright{} 1996 The American Physical Society.

ON THE EPSTEIN ZETA FUNCTION

The Epstein zeta function $Z(s)$ is defined for Re$s > 1$ by \[Z(s)=\sum_{m,n=-\infty, (m, n)\neq (0, 0)}^\infty \frac{1}{(am^2−bnm+cn^2)^s}\] where $a$, $b$, $c$ are real numbers with $a>0$ and $b^2 - 4ac<0$. $Z(s)$ can be continued analytically to the whole complex plane except for a simple pole at $s =1$. Simple proofs of the functional equation and of the Kronecker "Grenz-formel" for $Z (s)$ are given. The value of $Z(k)$($k =2, 3, \cdots$) is determined in terms of infinite series of the form \[\sum_{n=1}^\infty\frac{\cot^r n\pi\tau}{n^{2k-1}} (r=1, 2, \cdots, k)\] where $\tau=(b+\sqrt{b^2-4ac})/2a$.

Epstein zeta function and heavy-quark potential at finite temperature from Nambu-Goto string

From the Nambu-Goto string model we have calculated the static quark-antiquark potential at finite temperature. The Epstein zeta function is used for calculating the trace of the operator. The numerical results are given which suggest the quarkonium can be deconfined at finite temperature.

On a divisor problem related to the Epstein zeta-function

On a divisor problem related to the Epstein zeta-function

Zeros of quadratic zeta-functions on the critical line

In this paper, we consider the zeros of quadratic zeta-functions on the critical line. We begin by explaining the term quadratic zeta-functions. By this, we mean either the Epstein zeta-function associated with a positive definite binary quadratic form or the zeta-function of an ideal class in a quadratic field. One common feature of these things is that each of them has a functional equation of certain type (see §2, in particular (2.9)). The main result of this paper is an analogue of (1.1). If 1/2+iγ∗ n (γ ∗ n ≥ 0) is the nth zero of any of the quadratic zeta-functions mentioned above, we prove

Statistical properties of the zeros of zeta functions-beyond the Riemann case

We investigate the statistical distribution of the zeros of Dirichlet L-functions both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes we show that the two-point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different L-functions are statistically independent. Applications of these results to Epstein's zeta functions are briefly discussed.

An extension of the Chowla-Selberg formula useful in quantizing with the Wheeler-De Witt equation

The two-dimensional inhomogeneous zeta-function series (with homogeneous part of the most general Epstein type): \[ \sum_{m,n \in \mbox{\bf Z}} (am^2+bmn+cn^2+q)^{-s}, \] is analytically continued in the variable $s$ by using zeta-function techniques. A simple formula is obtained, which extends the Chowla-Selberg formula to inhomogeneous Epstein zeta-functions. The new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the Wheeler-De Witt equation in 2+1 dimensional spacetime with the torus topology.

On the Epstein zeta function

In this paper we give an elementary proof on the range of convergence of Epstein zeta functions.

Number variance of the zeros of the Epstein zeta functions

Number variance of the zeros of the Epstein zeta functions

A study of the motion of zeros of the Epstein zeta function associated to m2 + y2n2 as y varies from 1 to

This paper is concerned with the special Epstein zeta function defined by ▪ for σ \\2>1 and by analytic continuation in the entire s-plane. A numerical study of its zeros with 0 < t < 18 is described and for a selection of values of y these zeros are listed. Most of them are on the critical line σ = 1 2 , but for y in certain intervals there also is a pair of zeros not on this line and in some cases exactly two such pairs are revealed. Our calculations are based on a new representation of ( y/π) s Γ( s) Z( s,y) by factorial series, which is derived in the paper and which makes numerical computation of zeros as functions of y feasible on a medium sized computer.

EPSTEIN-FUNCTION ANALYSIS OF THE CASIMIR EFFECT AT FINITE TEMPERATURE FOR MASSIVE FIELDS

The Casimir free energy for a massive bosonic field subject to Dirichlet boundary conditions on hypercuboids of arbitrary dimensions is evaluated in the form of power and exponential expansions for the high- and low-temperature limits, respectively, with the aid of both homogeneous and inhomogeneous multidimensional Epstein zeta functions.

Inhomogeneous multidimensional Epstein zeta functions

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.