Barnes Zeta Function Research Articles

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s \zeta_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbf Q(x)$, where $\zeta_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function in the case that $x$ is a positive rational number, involving the generalized Stieltjes constants and the values at positive integers of the Riemann zeta function. Our formulae also makes it possible to calculate an approximation in the case that $w_1, \ldots, w_N$ and $x$ are positive real numbers.

On values of the higher derivatives of the Barnes zeta function at non-positive integers

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Multivariable Faulhaber-type formulas and Barnes zeta functions

A multivariable generalization of Faulhaber's formula is demonstrated and, using Barnes zeta functions, a multivariable generalization of an asymptotic formula for sums of non-integral powers of integers is shown. We also give an asymptotic formula for Barnes zeta functions.

Representations for the derivative at zero and finite parts of the Barnes zeta function

ABSTRACTWe provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes gamma function, including the particular case also known as multiple gamma function.

Modified Nörlund polynomials

The modified Bernoulli numbers $$B_{n}^{*}$$ considered by Zagier are generalized to modified Norlund polynomials $${B_{n}^{(\ell )*}}$$ . For $$\ell \in \mathbb {N}$$ , an explicit expression for the generating function for these polynomials is obtained. Evaluations of some spectacular integrals involving Chebyshev polynomials and of a finite sum involving integrals of the Hurwitz zeta function are also obtained. New results about the $$\ell $$ -fold convolution of the square hyperbolic secant distribution are obtained, such as a differential-difference equation satisfied by a logarithmic moment and a closed-form expression in terms of the Barnes zeta function.

Relations for Bernoulli–Barnes numbers and Barnes zeta functions

The Barnes ζ-function is [Formula: see text] defined for [Formula: see text], Re (x) > 0, and Re (z) > n and continued meromorphically to ℂ. Specialized at negative integers -k, the Barnes ζ-function gives [Formula: see text] where Bk(x; a) is a Bernoulli–Barnes polynomial, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing Bk(0; a) gives the Bernoulli–Barnes numbers. We exhibit relations among Barnes ζ-functions, Bernoulli–Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

Analytic continuation of the doubly-periodic Barnes zeta function

The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorph...

Analytic continuation of the doubly-periodic Barnes zeta function

The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points.

Values of twisted Barnes zeta functions at negative integers

In this paper, we study analytical and arithmetical properties of the twisted zeta function \(\Gamma (s)^{ - 1} \int_0^\infty {e^{ - xt} t^{s - 1} } \prod\nolimits_{j = 1}^N {\frac{{a_j t - \log (w^a j)}} {{1 - w^{a_j } e^{a_j t} }}dt} \), where ℜ(s) > N, ℜ(x) > 0, w ∈ ℂ\{0}, N ∈ ℤ, and a 1, …, a N have positive real parts. These functions have many interesting properties. We prove a collection of fundamental identities satisfied by zeta functions of this kind. For instance, special values of these zeta functions are related to twisted Barnes numbers and polynomials. This gives us a new elementary approach to new and known results concerning the Barnes zeta functions. In particular, we derive some well-known results on the Hurwitz zeta functions.

BILATERAL ZETA FUNCTIONS AND THEIR APPLICATIONS

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversion formula of the Dedekind eta function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.

Spherical Casimir pistons

A piston is introduced into a spherical lune Casimir cavity turning it into two adjacent lunes separated by the (hemispherical) piston. On the basis of zeta-function regularization, the vacuum energy of the arrangement is finite for conformal propagation in spacetime. For even spheres this energy is independent of the angle of the lune. For odd dimensions it is shown that for all Neumann, or all Dirichlet, boundary conditions the piston is repelled or attracted by the nearest wall if d = 3, 7, … or if d = 1, 5, … , respectively. For hybrid N–D conditions these requirements are switched. If a mass is added, divergences arise which render the model suspect. The analysis, however, is relatively straightforward and involves the Barnes zeta function. The extension to finite temperatures is made and it is shown that for the 3, 7, … series of odd spheres, the repulsion by the walls continues but that, above a certain temperature, the free energy acquires two minima symmetrically placed about the midpoint.

Determinants and conformal anomalies of GJMS operators on spheres

The conformal anomalies and functional determinants of the Branson-GJMS operators, P2k, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k ⩽ d/2 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishes for odd d. Taking the mode system on the sphere as the union of Neumann and Dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non-existence of ‘super-critical’ operators, 2k > d for even d. Significant use is made of the Barnes zeta function. The results are given in terms of ratios of determinants of operators on a (d + 1)-dimensional bulk dual sphere. For odd dimensions, the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations, yielding an integral over a Plancherel measure. The N–D determinant ratio is also found explicitly for even dimensions. Ehrhart polynomials are encountered.

Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions

Dans ce papier nous étudions trois types nouveaux de sommes shiftées de Dedekind-Apostol-Rademacher. Les premières sommes sont écrites à l’aide des formes modulaires de Jacobi et les deuxièmes sont écrites en termes de valeurs de fonctions cotangentes et les troisièmes sont exprimées à l’aide de valeurs spéciales de fonctions zêta multiples de Barnes. Le résultat principal de cet article est de montrer une loi de réciprocité de Dedekind satisfaites par ces nouvelles sommes. Nos résultats recouvrent ceux de Hall-Wilson-Zagier sur les sommes classiques de Dedekind-Rademacher, ceux de Beck-Berndt-Dieter sur les sommes cotangentes et d’autres résultats obtenus par Ota et Nagasaka sur les sommes de Dedekind, attachées aux dérivées premières de fonctions zêta de Barnes.

Some analytic continuations of the Barnes zeta function in two and higher dimensions

Formulas for the analytic continuation of the Barnes zeta function, and some affine extensions thereof, in two and more dimensions, are constructed. The expressions are used to deal with determinants of multidimensional harmonic oscillators. An example is therewith obtained of the multiplicative anomaly (or defect), associated with the most common definition (due to Ray and Singer [D.B. Ray, Reidemeister torsion and the Laplacian on lens spaces, Adv. Math. 4 (1970) 109–126; D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145–201; D.B. Ray, I.M. Singer, Analytic torsion for complex manifolds, Ann. Math. 98 (1973) 154–177]) of determinant of a pseudodifferential operator admitting a zeta function [M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, in: Functional Analysis on the Eve of the 21st Century, vol. 1, 1995, pp. 173–197].

Casimir energies for spherically symmetric cavities

A general calculation of Casimir energies --in an arbitrary number of dimensions-- for massless quantized fields in spherically symmetric cavities is carried out. All the most common situations, including scalar and spinor fields, the electromagnetic field, and various boundary conditions are treated with care. The final results are given as analytical (closed) expressions in terms of Barnes zeta functions. A direct, straightforward numerical evaluation of the formulas is then performed, which yields highly accurate numbers of, in principle, arbitrarily good precision.