The variation of Barnes and Bessel zeta functions

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.