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The variation of Barnes and Bessel zeta functions

Clara L. Aldana, Klaus Kirsten, Julie Rowlett

TL;DR

This work analyzes how two fundamental families of zeta functions, the Barnes zeta functions and the Bessel zeta functions, vary with a key defining parameter. It derives the derivative at zero for the Barnes family and its parameter-variation formula, including a novel closed form when the parameter is an integer. For the Bessel zeta family, it computes $\xi_0'(0)$ and relates $\xi_c'(0)$ to Barnes data, providing two independent methods to obtain the parameter variation and proving their equivalence through a network of identities involving elementary and special functions. The results connect to zeta-regularized determinants of Laplacians on cones and circular sectors, offering tools and identities with potential applications in spectral geometry and mathematical physics.

Abstract

We consider the variation of two fundamental types of zeta functions that arise in the study of both physical and analytical problems in geometric settings involving conical singularities. These are the Barnes zeta functions and the Bessel zeta functions. Although the series used to define them do not converge at zero, using methods of complex analysis we are able to calculate the derivatives of these zeta functions at zero. These zeta functions depend critically on a certain parameter, and we calculate the variation of these derivatives with respect to the parameter. For integer values of the parameter, we obtain a new expression for the variation of the Barnes zeta function with respect to the parameter in terms of special functions. For the Bessel zeta functions, we obtain two different expressions for the variation via two independent methods. Of course, the expressions should be equal, and we verify this by demonstrating several identities for both special and elementary functions. We encountered these zeta functions while working with determinants of Laplace operators on cones and angular sectors.

The variation of Barnes and Bessel zeta functions

TL;DR

This work analyzes how two fundamental families of zeta functions, the Barnes zeta functions and the Bessel zeta functions, vary with a key defining parameter. It derives the derivative at zero for the Barnes family and its parameter-variation formula, including a novel closed form when the parameter is an integer. For the Bessel zeta family, it computes and relates to Barnes data, providing two independent methods to obtain the parameter variation and proving their equivalence through a network of identities involving elementary and special functions. The results connect to zeta-regularized determinants of Laplacians on cones and circular sectors, offering tools and identities with potential applications in spectral geometry and mathematical physics.

Abstract

We consider the variation of two fundamental types of zeta functions that arise in the study of both physical and analytical problems in geometric settings involving conical singularities. These are the Barnes zeta functions and the Bessel zeta functions. Although the series used to define them do not converge at zero, using methods of complex analysis we are able to calculate the derivatives of these zeta functions at zero. These zeta functions depend critically on a certain parameter, and we calculate the variation of these derivatives with respect to the parameter. For integer values of the parameter, we obtain a new expression for the variation of the Barnes zeta function with respect to the parameter in terms of special functions. For the Bessel zeta functions, we obtain two different expressions for the variation via two independent methods. Of course, the expressions should be equal, and we verify this by demonstrating several identities for both special and elementary functions. We encountered these zeta functions while working with determinants of Laplace operators on cones and angular sectors.
Paper Structure (8 sections, 12 theorems, 246 equations, 2 figures)

This paper contains 8 sections, 12 theorems, 246 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. The Barnes zeta function
  4. The variation of the Barnes zeta function when the parameter is a natural number
  5. The Bessel zeta functions
  6. The variation of the Bessel zeta function when the parameter is a natural number
  7. The variation of the Bessel zeta function when the parameter is not a natural number
  8. Concluding remarks

Key Result

Theorem 1.1

Let $\zeta_c(z)$ be defined as in eq:def_zetac. Then the derivative with respect to $c$ of $\zeta_c'(0)$ is Here $\psi(x) = \frac{d}{dx} \log \Gamma(x)$ is the so-called di-Gamma function. If $c=j>1$ is an integer, then this expression is equal to

Figures (2)

  • Figure 1: The contour of integration, $\Gamma_R$.
  • Figure 2: The contour of integration, $\Gamma_{R, \epsilon}$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • ...and 17 more