The Barnes zeta-function, sphere determinants and Glaisher- Kinkelin-Bendersky constants

TL;DR

This paper extends summation relations to Barnes zeta-functions, introduces a Kaluza--Klein approach for determinants, and generalizes constants, improving the understanding of sphere determinants and special functions.

Contribution

It develops a Kaluza--Klein technique linking determinants to Glaisher-Kinkelin-Bendersky constants and extends these concepts to general zeta-functions.

Findings

Extended relations to Barnes zeta-functions.

Introduced a determinant interpretation of constants.

Enhanced methods for calculating sphere determinants.

Abstract

Summations and relations involving the Hurwitz and Riemann zeta-functions are extended first to Barnes zeta-functions and then to zeta-functions of general type. The analysis is motivated by the evaluation of determinants on spheres which are treated both by a direct expansion method and by regularised sums. Comments on existing calculations are made. A Kaluza--Klein technique is introduced providing a determinant interpretation of the Glaisher-Kinkelin-Bendersky constants which are then generalised to arbitrary zeta-functions. This technique allows an improved treatment of sphere determinants.